,„,.3^So^,«aMun3U 


A                        LIBRARY                               ^^ 

\                              "^                               j^^ 

\  Purdue  UNiVERSips^ 

AGRICUi}tURAL    EXPERlM,g1<T    STATION 

^eceS^my^or,..81.98.__. 

CTass .^^.-                       >w                        JSooA: 

^                         N. 

sawwi^f'nr 


Digitized  by  the  Internet  Archive 

in  2009  with  funding  from 

Boston  Library  Consortium  Member  Libraries 


http://www.archive.org/details/manualoftopograpOOgann 


LIBRARY  CATALOGUE  SLIPS. 

United  States.     Department  of  the  interior.     (  U.  S.  (/eologieal  siirveii.) 

Department  of  the  interior  \  —  \  Monographs  |  of  the  |  United 
States  geological  survey  |  Volume  XXII  |  [Seal  of  the  depart- 
ment] 1  Washington  |  government  printing  offlce  |  1893 

Second  title:  United  States  gecdogical  survey  |  J.  W.  Powell 
director  |  —  |  A  manual  |  of  |  topographic  methods  |  by  |  Henry 
Gannett  |  chief  topographer  |  f  Vignette]  | 

Washington  |  government  printing  office  |  1893 

4°.     XIV,  300  pp.  18  pi. 


Gannett  (Henry). 

United  States  geological  survey  1  .1.  W.  Powell  director  |  —  | 
A  manual  |  of  |  topographic  methods  |  by  |  Henry  Gannett  |  chief 
topographer  |  [Vignette]  | 

Washington  |  government  printing  ofHcc  |  1893 

40.     XIV,  300  pp.  18  pi. 

[United  States.    De-pai-tmenl  ■../'   the   intiTiuy.     (T.  .S'.   tjeologieul  survey). 
Mono^rapli  SXlI.j 


United  States  geological  survey  |  .1.  W.  Powell  director  |  —  | 
A  manual  |  of  |  topographic  methods  |  by  |  Henry  Gannett  |  chief 
topographer  |  [Vigucttc]  | 

Washington  |  govcrniofut  printing  office  |  1893 

4°.    XIV,  300  pp.  18  pi. 

[Uniteu  States.    Departmnit  uf   the  interior.     iU.   *.  geoloyical  survey. 
Monograpli  XXII.] 


A^DVERTISE]Vd:ENT. 

[Monograph  XXII.  ] 


The  puhlicatinns  of  thp  TTnUert  Stntes  Geolo<;iciil  Survey  are  issued  iu  aocurdauce  with  the  statute 
approved  iMareh  :-!,  1871t,  wlii.li  declaifs  tliat— 

"The  publi<'a  tioiis  of  the  Geolof;iiMl  Survey  shall  cousist  of  the  auuuiil  r<'i)(irt  of  operations,  geo- 
logical aud  ecououiic  maps  iUustratiuj;-  the  resources  and  classilication  of  the  lands,  an<l  re|iorts  upon 
general  and  economic  f;i'olo,uy  and  paleontology.  Tlie  annual  rei>ort  of  operations  of  the  (ieologieal 
Survey  shall  aecompauy  the  a)uiual  report  of  the  Secretary  of  the  Interior.  All  sjiecial  nuauoirs  and 
reports  of  said  Survey  shall  be  issne,<l  in  uniform  quarto  series  if  deemed  necessary  liy  the  1  director,  Init 
otherwise  in  ordinary  octavos.  Three  thousand  copies  of  each  shall  hi>  pnMished  lorscieiitiHc  exclL-niues 
and  for  sale  at  the  price  of  publication;  and  all  literary  and  cartoniajdiic  niateiials  icreived  in  excliange 
shall  be  the  property  of  the  United  States  and  form  a  i>art  of  the  liiuary  ot  tin-  organization  ;  And  the 
money  resulting  from  the  sale  of  such  publications  shall  be  covered  into  the  Treasury  of  the  United 
States." 

The  following  joint  resolution,  referring  to  all  government  publications,  was  i)assed  by  Congress 
July  7,  1882 : 

"That  whenever  any  document  or  report  shall  be  ordered  printed  by  Congress,  there  shall  be 
)u-iuted,  in  addition  to  the  number  iu  eaih  case  stated,  the  '  usual  number '  (1,900)  of  copies  for  binding 
and  distribution  among  those  entitled  to  receive  them." 

Except  in  those  cases  in  which  au  extra  number  of  any  publication  has  been  supplied  to  the  Sur- 
vey by  special  resolution  of  Congress  or  has  been  ordered  by  the  Secretary  of  the  Interior,  this  ottice 
has  no  copies  for  gratuitous  distribution. 

ANNUAL  REPORTS. 

I.  First  Annual  Keport  of  the  Uuited  States  Geological  Survey,  by  Clarence  King.  1880.  8'^.  79 
pp.     1  map. — A  preliuiinarv  report  ilescribing  plan  of  organization  and  publications. 

II.  Second  Aiiuual  Kriioit  of  the  United  States  Geological  Survey,  1880-'81,  by  J.  W.  Powell. 

1882.  8°.     Iv,  .58Spp,     til' pi.     1  niaji. 

III.  Third  Annual  Kcpoit  of  tlic  United  States  Geological  Survey,  1881-'82,  by  J.  W.  Powell. 

1883.  8°.     xviii,5134  pp.     1)7  |d.  and  maps. 

IV.  Fourth  Annual  Report  of  the  United  States  Geological  Survey,  188L'-'88,  by  J.  W.  Powell. 

1884.  8°.     xxsii,  173  pp.     85  pi.  and  ina|is. 

V.  Fifth   Annual    Report  of  tlie  United   .States   Geological   Survey,  18S3-'81,  by  .1.  W.  Powell. 

1885.  8°.     xxxvi,  469pi).     .58  pi.  and  nuips. 

VI.  Sixth  Annual  Report  of  the  Uniteil  States  Geological  Survey,  1884-'8.5,  by  .J.  W.  Powell. 
1885.     8'^.     xxix,  570  pp.     65  pi.  and  maps. 

VII.  Seventh  Annual  Report  of  the  United  States  Geological  Survey,  1885-'86,  by  J.  W.  Powell. 

1888.  8°.     XX,  656  pp.     71  pi.  and  maps. 

VIII.  Eighth  Annual  Report  of  the  United  States  Geological- Survey,  1886-'87,  by  J.  W.  Powell 

1889.  8^'.     2v.     xix,  474,  xii  pp.     53  pi.  and  maps;  1  p.  1.     475-1063  pp.     54-76  pi.  and  maps. 

IX.  Ninth  Annual  Report  of  the  United  States  Geological  Survey,  1887-'88,  by  ,J.  W.  Powell. 

1889.  8'^.     xiii,  717  pp.     88  pi.  and  maps. 

X.  Tenth  Annual   Report  of  the  United  States  Geological   Survey,  1888-'89,  by  .J.  W.  Powell. 

1890.  8°.     2v.     XV,  774  pp.     98  pi.  and  maps;  viii,  123  pp. 

XI.  Eleventh  Annual  Report  of  the  United  States  Geological  Survey,  1889-'9(),  by  ,J.  W.  Powell. 

1891.  8--".     2v.     XV,  7.57  pp.     66  pi.  and  maps;  ix,  351  pp.     .30  pi.  and  maps. 

XII.  Twelfth  Annual  Report  of  the  United  States  Geological  Survey,  1890-'91,  by  J.  VV.  Powell. 
1891.     8°.    2v.     xiii,  675  pp.     53  pi.  and  maps;  xviii,  .576  pp.  .  146  jd.  and  maps. 

XIII.  Thirteenth  Annual  Report  of  the  United  States  Geological  Survey,  1891-'92,  by  .1.  W. 
Powell,  1893.     8°.     3  v. 


II  ADVKK'l'lSKMENT. 

MONOGRAPHS. 

I.  Lake  Rouneville,  liy  Grove  Kail  fiilbort.     1890.     4^^.     xx,  438  pp.     51  pi.     1  map.     Price  $1. .50. 

II.  Tertiary  History  of  the  Grand  ( 'anon  District,  with  atlas,  liy  t'larc.nce  IJ.  Duttoii,  Capt.,  U.  S.  A. 
1882.     4^'.     xiv,  L'til  pp.     'i'2  pi.  and  atlas  of  21  sheets  folio.     Price  ^ilO.OO. 

III.  Geology  of  the  Comstock  Lode  aud  the  Washoe  District,  with  atlas,  liy  (Jeorgc  F.  Keeker. 
1882.     4-\     XV,  422  pp.     7  pi.  and  atlas  of  21  sheets  folio.     Price  $11.00. 

IV.  Comstock  Mining  aud  Miners,  by  Eliot  Lord.     1883.     4"^.     xiv,  451  pp.     3  pi.     Price  $1.50. 

V.  The  t'oppcr-Beari'ng  Rocks  of  Lake  Superior,  by  Roland  Duer  Irving.  1883.  4  '.  xvi,  llil 
pp.     15  1.     29  pi.  aud  maps.  "Price  $1.85. 

VI.  Coutributious  to  tlie  Knowledge  of  the  Older  Mesozoic  Flora  of  Virginia,  by  William  Morris 
Fontaine.     1883.     4-'.     xi,  144  pp.     54 1.     .54  pi.     Price  $1.05. 

VII.  Silver-Lead  Deposits  of  Eureka,  Nevada,  by  Joseph  Story  Curtis.  1884.  4'.  xiii,  200  pp. 
16  pi.     Price  $1.20. 

Vm.  Paleontology  of  the  Eureka  District,  by  t'harles  Do.dittle  Walcott.  1884.  4'-.  xiii,  298 
pp.     24  1.     24  pi.     Price' $1.10. 

IX.  Brachiopoda  and  LamellOiranchiata  of  the  Raritau  Clays  and  Grecusand  Marls  of  New 
Jersey,  by  Robert  P.  AVhittield.     1885.     4-.     xx,  338  pp.     35  pi.     1  map.     Price  $1.15. 

X.  Dinocerata.  A  Monograph  of  an  Extinct  Order  of  Gigantic  Mammals,  by  Othuiel  Charles 
Marsh.     1886.     4>-\     xviii,  243  pp.     56  1.     .56  pi.     Price  $2.70. 

XI.  Geological  History  of  Lalce  Lahontaii,  a  yuateraary  Lake  of  Northwestern  Nevada,  by 
Israel  Cook  Russell.     1885.  '  I'-",     xiv,  288  pp.     46  pi.  and  maps."    Price  $1.75. 

XII.  Geology  and  Mining  Industry  of  Jjeadville.  Coldvado,  with  atlas,  by  Samuel  Franklin  Em- 
mons.    1886.     4^."  xxix,  770  pp.     45  pl.'aud  atlas  ol'3r>  slnrts  lolin.     Price  $8.40. 

XIII.  Geologv  of  the  Quicksilver  Dejiosits  uf  thi-  l^nilir  sl,,|ie,  with  atlas,  by  George  F.  Becker. 
1888.     4^.     xix,  486  pp.     7  pi.  and  atlas  of  14  sheets  loli...     i'licc  .$2.00. 

XIV.  Fossil  Fishes  and  Fossil  Plants  of  the  Triassic  Rocks  of  New  Jersey  and  the  Connecticut 
Valley,  by  John  S.  Newberry.     1888.     4°.'    xiv,  152  pp,     26  pi.     Price  $1.00. 

XV.  The  Potomac  or  Younger  Mesozoic  Flora,  by  William  Morris  Fcmtaine.  1889.  -l^.  'xiv, 
377  pp.     180  pi.     Text  and  plates  bmind  separately.     Price  $2..50. 

XVI.  The  Paleozoic  Fishes  of  North  America,  by  John  Strong  Newberry.  1889.  4-.  340  pp. 
53  pl.     Price  $1.00. 

XVII.  The  Flora  of  the  Dakota  Group,  a  posthumous  work,  by  Leo  Lesquereux.  Edited  by  F. 
H.  Knowlton.     1891.     4^.     400  pp.     66  ]d.     Price  $1.10. 

XVIII.  Gasteropoda  aud  Cephalopoda  of  the  Raritan  Clays  and  Greeusand  Marls  of  New  Jersev, 
by  RobertP.  Whittisld.     1891.     4-\     402  pp.     50  pi.     Price  $1.00. 

XIX.  The  Penokee  Iron-Bearing  Series  of  Northern  Wisconsin  aud  Michigan,  by  Kolanil  D. 
Irving  and  C.  R.  Van  Rise.     1892.     4°.     xix,  534  pp.     Price  $1.70. 

XX.  Geology  of  tlie  Eureka  District,  Nevada,  with  an  atlas,  by  Arnold  Hague.  1892.  4'^'.  xvii. 
419  pp.     8  1)1.     Price  $5.25. 

XXI.  The  Tertiarv  Rhvnchophorous  Coleoptera  of  the  United  States,  by  Samuel  Hubbard  Scud- 
der.     1893.     4°.     xi.  206  pp.  12  pl.     Price  90  cents. 

XXII.  A  Manual  of  Topographic  Methods,  bv  Henry  Gannett,  chief  toiiographer.  1893.  4- . 
XIV.  300  pp.  18  pl.   Price  $1.00. 

In  press: 

XXIII.  Geology  of  the  Green  Mountains  in  Massachusetts,  bv  Raphael  Pnnipellv,  T.  Nelson  Dalei 
and  .T.  E.  Wolff. 

In  iireparation ; 

— Mollusca  and  Crustacea  of  the  Miocene  Formations  of  New  Jersey,  by  R.  P.  Whitfield. 

— SauTopoda,  by  0.  C.  Marsh. 

— Stegosauria,  by  O.  C.  Marsh. 

— Brontotheridte,  by  O.  C.  Marsh. 

— Rejjort  on  the  Denver  Coal  Basin,  by  S.  F.  Emmons. 

— Report  on  Silver  Cliff  and  Ten-Mile  Mining  Districts.  Colorado,  by  S.  F.  Emmons. 

— The  Glacial  Lake  Agassiz,  by  Warren  Upluim. 

BULLETINS. 

1.  On  Hypersthene-Andesite  aud  on  Tricliuic  Pyroxene  in  Augitic  Rocks,  by  Whitman  Cross, 
with  a  Geological  Sketch  of  Buffalo  Peaks,  Colorado,  by  S,  F.  Emmons.  1883.  8'^.  42  pp.  2  i)l. 
Price  10  cents. 

2.  Gold  and  Silver  Conversion  Tables,  gjvnig  tlie  coining  values  of  troy  ounces  of  fine  metal, -etc., 
computed  by  Albert  Williams,  jr.     1883.     8*^.     8  p]i.     Price  5  cents. 

3.  On  the  Fossil  Faunas  of  the  Upper  Devonian,  .along  the  meridian  of  76°  30',  from  Tompkins 
County,  N.  Y.,  to  Bradford  County,  Pa.,  by  Henry  S.  Williams.     1884.     8'^'.     36  pp.     Price  5  cents 

4.  On  Mesozoic  Fossils,  by  Charles  A.  White.     1884.     8-.     36  pp.     9  pl.     Price  5  cents. 

5.  A  DictionaTy  of  Altitudes  in  the  United  States,  compiled  by  Henry  Gannett.  1884.  8°.  325 
pp.     Price  20  cents. 

6.  Elevations  in  the  Dominion  of  Canada,  by  J.  W.  Spencer.     1884.     8°.     43  pp.     Price  5  cents. 

7.  Mapoteca  Geologica  Americana.  A  Catalogue  of  Geological  Maps  of  America  (North  and 
South),  1752-1881,  in  geographic  and  chronologic  order,  by  Jules  Marcou  and  John  Belknap  Marcou. 
1884.    8°.    184  pp.    Price  10  cents. 


ADVERTISEMENT.  IH 

8.  On  Seooudiiry  Enlargemeuts  of  Mineral  Fragments  iu  Certain  Rcicks,  by  E.  D.  Irving  and  C. 
E.  VaiiHise.      I.SSI,    's  .     5li  pp.      ti  pi.      Price  10  cents. 

9.  A  Eeport  ol' W(iik  ihnw  in  tlip  Wasliiiigtim  Laboratory  dnring  the  fiscal  year  18!^3-'84.  F.  W. 
Clarke,  chief  chemist;  T.  M.  Chatanl,  assistaiit  chemist.      1884.      8.      40  pp.      Price  5  cents. 

10.  On  the  Cambrian  Fauua.s  of  Nnrtli  America.  Preliminary  studies,  by  Charles  Doolittle  Wal- 
cott.      1884.      8°.      74  pp.      10  pi.      Price  5  cents. 

11.  On  the  Quaternary  and  Eecent  Mollusca  of  the  Great  Basin;  with  Descriptions  of  New 
Forms,  by  E.  Ellsworth  Call'  Introduced  by  a  sketch  of  the  Quaternary  Lakes  of  the  Great  Basin, 
by  G.  K.  Gilbert.      1884.     8°.     66  pp.     6  pi.  '  Price  5  cents. 

12.  A  Crystallographic  Study  of  the  Thinolite  of  Lake  Lahontan,  by  Edward  S.  Dana.  1884.  8'-'. 
34  pp.  '  3  pi.     Price  5  cents. 

13.  Boundaries  of  the  United  States  and  of  tlie  several  States  and  Tei'ritories,  with  a  Historical 
Sketch  of  the  Territorial  Changes,  by  Henry  Gauuett.     1885.    »-.     135  pp.     Price  10  cents. 

14.  The  El«!Ctr)cal  and  Magnetic  Properties  of  the  Iron-Carburets,  by  Carl  Barns  and  Vincent 
Strouhal.     1885.     8'^.     238  pp.     Price  15  cents. 

15.  On  the  Mesozoic  and  Cenozoic  Paleontology  of  California,  by  Charles  A.  AAHiite.  1885.  8°. 
33  pp.     Price  5  cents. 

16.  On  theHigherDevonianFanuasof  Ontario  County,  New  York,  by  John  M.  Clarke.  1885.  8°. 
86  pp.     3  pi.     Price  5  ceuts. 

17.  On  the  Development  of  Crystallization  in  the  Igneous  Eocks  of  Washoe,  Nevada,  with  Notes 
on  the  Geology  of  the  District,  by  Arnold  Hague  and  Joseph  P.  Iddings.  1885.  8°.  44  pp.  Price  5 
cents. 

18.  On  Marine  Eocene,  Fresh-water  Miocene,  and  other  Fossil  Mollusca  of  Western  North  America, 
by  Charles  A.  White.     1885.    8*^.     26  pp.     3  pi.     Price  5  cents. 

19.  Notes  on  the  Stratigraphy  of  California,  by  George  F.Becker.    1885.   8^^.   28pp.   Price5ceuts. 

20.  Contributions  to  the  Miiieralogv  of  the  Rocky  Jlountains,  by  Whitman  Cross  and  W.  F.  Hille- 
brand.     1885.     8'-.     114  pp.     1  pi.     Price"  10  cents. 

21.  The  Lignites  of  the  Great  Siovix  Eeservation.  A  Eeport  on  the  Eegion  between  the  Grand 
and  Moreau  Rivers,  Dakota,  by  Bailey  Willis.     1885.     8'-.     16  pp.     5  pi.     Price  5  cents. 

22.  On  New  Cretaceous  Fossils  from  California,  by  C'harles  A.  White.  1885.  8-^.  25  pp.  5  pi. 
Price  5  cents. 

23.  Observations  on  the  .Junction  between  the  Eastern  Sandstone  and  the  Keweenaw  Series  on 
Keweenaw  Point,  Lake  Superior,  by  R.  D.  Irving  and  T.  C.  Chamberlin.  1885.  8'^.  124  pp.  17  pi. 
Price  15  cents. 

24.  List  of  Marine  Mollusca,  comprising  the  Quaternary  fossils  and  recent  forms  from  American 
Localities  between  Cape  Hatteras  and  Cape  Roque,  including  the  Berjnudas,  by  William  Healey  Dall. 
1885.     8°.     336  pp.     Price  25  cents. 

25.  The  Present  Technical  Condition  of  the  Steel  Industry  of  the  United  States,  by  Phineas 
Barnes.     1885.     8°.     85  pp.     Price  10  cents. 

26.  Copper  Smelting,  by  Henry  M.  Howe.     1885.     8^^.     107  pp.     Price  10  cents. 

27.  Report  of  work  done  in  the  Division  of  Chemistry  and  Physics,  mainly  during  the  fiscal  year 
1884-'85.     1886.     8°.     80  pp.     Price  10  cents. 

28.  The  Gabbros  and  Associated  Hornblende  Rocks  occurring  iu  the  Neighborhood  of  Baltimore, 
Md.,  by  George  Huntington  Williams.     1886.     8^.     78  pp.     4  pi.    Price  10  cents. 

29.  On  the  Fresh-water  Invertebrates  of  the  North  American  Jurassic,  by  Chiirles  A.  White.  1886. 
8^'.    41  pp.     4  pi.     Price  5  cents. 

30.  Second  Contribution  to  the  Studies  on  the  Cambrian  Faunas  of  North  America,  by  Charles 
Doolittle  Walcott.     1886.     8"^.     369  pp.     33  pi.     Price  25  cents. 

31.  Systematic  Review  of  our  Present  Knowledge  of  Fossil  Insects,  including  Myriapods  and 
Arachnids,  by  Samuel  Hubbard  Scudder.     1886.     8°.     128  pp.     Price  15  cents. 

32.  Lists  and  Analyses  of  the  Mineral  Springs  of  the  United  States;  a  Preliminary  Study,  by 
Albert  C.  Peale.     1886.     8°.     235  pp.     Price  20  ^■ents. 

33.  Notes  on  the  Geolo'iy  of  Northern  California,  by  J.  S.Di.ler.     1886.    8°.    23  pp.    Price  5  cents. 

34.  On  the  relation  of  the  Laramie  Molhiscan  Fauna  to  that  of  the  succeeding  Fresh-water  Eocene 
and  other  groups,  by  Charles  A.  AVhite.     1886.     8^.     .54  pp.     5  pi.     Price  10  cents. 

35.  Physical  Properties  of  the  Iron-Carburets,  by  Carl  Barns  and  Vincent  Strouhal.  1886.  8°. 
62  pp.     Price  10  cents. 

36.  SubsidenceofFineSolidParticlesiuLiquid»,bvCarlBarus.    1886.    8°.    58pp.    PricelOceuts. 

37.  Types  of  the  Laramie  Flora,  hv  Lester  F.  Ward.     1887.     8°.     354  pp.     57  pi.     Price  25  cents. 

38.  PeridotiteofEUiottCounty,  Kentucky,  by  J.  S.Diller.     1887.     8^.    31pp.    Ipl.    Price5cents. 

39.  The  Upper  Beaches  and  Deltas  of  the  Glacial  Lake  Agassiz,  by  Warren  Upham.  1887.  8". 
84  pp.     1  pi.     Price  10  cents. 

40.  Changes  iu  River  Courses  in  Washington  Territory  due  to  Glaciation,  by  Bailey  AVillis.  1887. 
8°.     10  pj).     4  pi.     Price  5  ceuts. 

41.  On  the  Fossil  Faunas  of  the  Upper  Devonian— the  Genesee  Section,  New  York,  by  Henry  S. 
Williams.     1887.     8°.     121  pp.     4  pi.     Price  15  cents. 

42.  Report  of  work  done  in  the  Division  of  Chemistry  and  Physics,  mainly  during  the  fiscal  year 
1885-'86.     F.W.Clarke,  chief  chemist.     1887.     8".     1.52  pp.     Ipl.     Price  15  cents. 

43.  Tertiary  and  Cretaceous  Strata  of  the  Tuscaloosa,  Tombigbee,  and  Alabama  Rivers,  l>y  Eugene 
A.  Smifh  and  Lawrence  C.  Johnson.     1887.    8".     189  pp.     21  pi.    Price  15  cents. 


IV  .  ADVEKTISEMKNT. 

U.  Bihliogi-iiiUy  of  North  Aiiiericau  (ieDlogy  lor  1S86,  by  NolsoM  II.  Oiirtoii.  1887.  8^'.  :«  pj). 
Price  5  cents. 

45.  The  Present  Condition  o!'  Knowledge  of  the  ( icologv  iif  Texns.  )>y  lvob.-r(-  T.  Hill.  1887.  8  , 
94  P11.     Price  10  cents. 

4ti.  Nature  and  Origin  of  Deposits  of  Phosplialc  of  Liiac.  l>v  K.  A.  V.  Peni'ose,  jr.,  with  an  Intro- 
duction by  N.  S.  Shalcr.     1888.     8«.     143  pp.     Price  If.  cents. 

47.  Analyses  of  Waters  of  tbe  Yellowstone  Natiinial  Paik,  wilh  .ui  Account  of  tlie  Metlnxls  of 
Analysis  employed,  by  Frank  Austin  Goo'ch  and  James  Edwaril  Whitlicld.  1888.  S'^.  84  i)p.  I'rice 
10  cents. 

48.  On  the  Form  and  Position  of  the  Sea  Level,  by  Kobcrl  Simpson  Woodward.  1888.  8'-'.  88 
pp.     Price  10  cents. 

49.  Latitudes  and  Longitudes  of  t'ertain  Points  in  Missouri,  Kansas,  and  New  Mexico,  by  Kobert 
Simpson  Woodward.     1889.  ''8'-\     133  pp.     Price  15  cents. 

50.  Formulas  and  Tables  to  Facilitate  the  Construction  and  Use  of  Maps,  by  Robert  Simpson 
Woodward.     1889.     8'^.     124  pp.     Price  15  cents. 

51.  On  Invertebrate  Fossils  from  the  Pacific  Coast,  by  Charles  Abiathar  White.  1889.  8  .  102 
pp.     14  pi.     Price  15  cents. 

52.  Subaerial  Decay  of  Rocks  and  Origin  of  the  Red  Color  of  Certain  Formations,  by  Israel 
Cook  Russell.     1889.     8°.'   65  pp.     5  pP    Price  10  cents. 

53.  The  Geology  of  Nantucket,  by  Nathaniel  Southgate  Shaler.  1889.  8*^.  55  pp.  10  pi.  Price 
10  cents. 

54.  On  the  Thernio-Electric  Measurement  of  High  Temperatures,  by  Carl  Barns.  1889.  8°. 
313  pp.,  incl.  1  pi.     11  pi.     Price  25  cents. 

55.  Report  of  work  done  in  the  Division  of  Chemistry  and  Physics,  mainly  during  the  tiscal 
year  1886-'87.     Frank  Wigglcsworth  Clarke,  chief  chemist.     1889.     8".     96  pp.     Price  10  cents. 

56.  Fossil  \\'ood  and^Liguite  of  the  Potomac  Formation,  by  Frank  Hall  Knowlton.  1889.  8"^. 
72  pp.     7  pi.     Price  10  cents. 

57.  A  Geological  Recouuoissauce  in  .Southwestern  Kansas,  by  Robert  Hay.  1890.  ■  8^\  49  pp. 
2  pi.     Price  5  cents. 

58.  The  Glacial  Boundary  in  Western  Peuusylvauia,  Ohio,  Kentucky,  Indiana,  and  Illinois,  by 
George  Frederick  Wright,  with  an  introduction  by  Thomas  Chrowder  Chamberlin.  1890.  8°.  112 
pp.  incl.  1  pi.     8  pi.     Price  15  cents. 

59.  The  Gabbros  and  Associated  Rocks.in  Delaware,  by  Frederick  D.  Chester.  1890.  8*^'.  45 
pp.     1  pi.     Price  10  cents. 

60.  Report  of  work  done  in  the  Division  of  Chemistry  and  Physics,  mainly  during  the  tiscal 
year  1887-'88.     F.  W.  Clarke,  chief  chemist.     1890.     8*^.     174  pp.     Price  15  cents. 

61.  Contributions  to  the  Mineralogy  of  the  Pacific  Coast,  by  William  Harlow  Melville  and  Wa4- 
demar  Lindgren.     1890.     8°.     40  pp.     3  pi.     Price  5  cents. 

62.  The  Greenstone  Schist  Areas  of  the  Menominee  and  Marquette  Regions  of  Michigan,  a  eou- 
tribution  to  the  snbjei  t  of  dynamic  metamorphism  in  eruptive  rocks,  by  George  Huntington  Williams, 
with  an  introduction  by  Roland  Duer  Irving.     1890.     8°.     241  pp.     16  pi.     Price  30  cents. 

63.  A  BibliogTaphy  of  Paleozoic  Crn.stacea  from  1698  to  1889,  including  a  list  of  North  Amer- 
ican species  and  a  systematic  arrangement  of  genera,  by  Anthony  W.  Vogdes.  1890.  8"=.  177  pp. 
Price  15  cents. 

64.  A  Report  of  work  done  in  the  Division  of  Chemistry  and  Pliysics,  mainly  during  the  fiscal 
year  1888-'89.     F.  W.  Clarke,  chief  diemist.     1890.     8°.     60  pp.     Price  10  cents. 

65.  Stratigraphy  of  the  Bituminous  Coal  Field  of  Pennsylvania,  Ohio,  and  West  Virginia,  l)y 
Israel  C.  White.     1891.     8°.     212  pp.     11  pi.     Price  20  cents. 

66.  On  a  Group  of  Volcanic  Rocks  from  the  Tewan  Mountains,  New  Jlesico,  and  on  the  occur- 
rence of  Primary  Qnartz  in  certain  Basalts,  by  Joseph  Paxson  Iddings.  1890.  8°.  34  pp.  Price  5 
cents. 

67.  The  relations  of  the  Traps  of  the  Newark  System  in  the  New  Jersey  Region,  by  Nelson 
Horatio  Dartou.     1890.     8-^.     82  pp.     Price  10  cents. 

68.  Earthquakes  in  California  in  1889,  by  James  Edward  Keeler.  1890.  8°.  25  pp.  Price  5 
cents. 

69.  A  Classed  and  Annotated  Biographv  of  Fossil  Insects,  by  Samuel  Howard  Scudder.  1890. 
8°.     101pp.     Price  15  cents. 

70.  A  Report  on  Astronomical  Work  of  1889  and  1890,  by  Robert  Simpson  Woodward.  1890.  8'-'. 
79  pp.     Price  10  cents. 

71.  Index  to  the  Known  Fossil  Insects  of  the  World,  including  Myriapods  and  Arachnids,  by 
Samuel  Hubbard  Scudder.     1891.     8°.     744  pji.     Price  50  cents. 

72.  Altitudes  between  Lake  Superior  and  the  Rooky  Mountains,  by  Warren  Fpliam.  1891.  8". 
229  pp.     Price  20  cents. 

73.  The  Viscosity  of  Solids,  by  Carl  Barns.     1891.     8^.     xii,  139  pp.     6  pi.     Price  15  cents. 

74.  The  Minerals  of  North  Carolina,  by  Frederick  Augustus  Genth.  1891.  8°.  119  pp.  Price 
15  cents. 

75.  Record  of  North  American  Geology  for  1887  to  1889,  inclusive,  by  Nelson  Horatio  Darton. 
1891.     8'^.     173  pp.     Price  15  cents. 

76.  A  Dictionary  of  Altitudes  in  the  United  States  (second  edition ).  compiled  by  Henry  Gannett, 
chief  topographer.     1891.     8^.     393  pp.     Price  25  cents. 


ADVERTISEMENT.  V 

77.  The  Texar.  Permian  and  its  Mesozoic  types  of  Fossils,  liy  Charles  A.  White.     1891.     .S-.     51 
pp.     4  pi.     Price  10  cents. 

78.  A  report  of  -n'ork  done  in  the  Division  of  Chemistry  and  Physics,  mainlv  duriii"-  the  liscal 
year  lS89-'90.     F.  W.  Clarke,  chief  chemist.     1891.     8'^.     131  pp.     Price  1.5  cents.   ' 

79.  A  Late  Volcanic  Eruption  in  Northern  California  and  its  peculiar  lava,  by  J*;  S.  Diller. 

80.  Correlation  papers — Devonian  and  Carboniferous,  by  Henry  Shaler  Williams.     1891.     8°. 
279  pp.     Price  20  cents. 

81.  Correlation  papers — C.imbviau,  by  Charles  Doolittle  Walcott.      1891.     8^.     547  pp.     3  pi. 
Price  25  cents. 

82.  Correlation  papers — Cretaceous,  by  Charles  A.  White.     1891.     8^.     273  pp.     3  pi.     Price  20 
cents. 

83.  Correlation  papers— Eocene,  by  AA'illiara  Bullock  Clark.     1891.     S°.     173  pp.     2  pi.     Price 
15  cents. 

84.  Correlation  papers— Neocene,  by  W.  H.  Dall  and  Q.  D.  Harris.  _  1892.     8^.     349  pp.     3  pi. 
Price  25  cents. 

85.  Correlation  papers^The  Newark  System,  by  l.srael  Cook  Russell.     1892.     8^.     344  pp.    13  pi. 
Price  25  cents. 

86.  Correlation  papers — Archean  and  Algonkian,  by  C.  E.  Van  Hise.     1892.     8^^.     .549  pp.     12  pi. 
Price  25  cents. 

90.  A  report  of  Tvork  done  in  the  Division  of  Chemistry  and  Phy.sics,  mainly  during  the  tisoal 
year  1890-'91.     F.  W.  Clarke,  chief  chemist.     1892.     8".     77  pp.'     Price  10  cents. 

91.  Record  of  North  American  Geology  for  1890,  by  Nelson  Horatio  Darton.     1891.     8'J.     88  pp. 
Price  10  cents. 

92.  The  Compressibility  of  Liquids,  by  Carl  Barus.     1892.    8°.     96  pp.    29  pi.     Price  10  cents. 

93.  Some  Insects  of  sjn-i-ial  iuterrst  fioui  Florissant,  Colorado,  and  other  points  in  the  Tertiaries 
of  Colorado  and  Utah,  by  Sauincl  Hubbard  Scudder.     1892.     8^.     35  ])p.     3  pi.     Price  5  cents. 

94.  The  Slechanism  of  >olid  Yi.scosity,  by  Carl  Barns.     1892.     8'^\     138  pp.     Price  15  cents. 

95.  Earthquakes  in  California  in  1890  and  1891,  by  Edward  Singleton  Holdeu.     1892.    8^.     31pp. 
Price  5  cents. 

96.  The  Volume  Thermodynamics  of  Liquids,  by  Carl  Barus.     1892.     8^.     100pp.     Price  10  cents. 

97.  The  Mesozoic  Echinodermata  of  the  United  States,  by  W.B.  Clark.    1893.    8".    207  pp.    iiOpl. 
Price  20  cents. 

98.  Flora  of   the  Outlying  Carboniferous  Basins  of  Southwestern   Missouri,  by  David  White. 
1893.    8^.     139  pp.     5  pi.     Price  15  cents. 

99.  Record  of  North  American  Geology  for  1891,  by  Nelson  Horatio  Darton.     1892.     8-.     73  pp. 
Price  10  cents. 

100.  Bibliography  and  Index  of  the  Publications  of  the  U.  S.  Geological  Survey,  1879-1892,  by 
Philip  Creveling  Warman.     1893.     8'^.     495  pp.     Price  25  cents. 

101.  Insect  fauna  of  the   Rhode   Island   Coal  Field,  by   Samuel   Hubbard   Scudder.     1893.     8-'. 
27  pp.     2  pi.     Price  5  cents. 

103.  High  Temperature  Work  in  Igneous  Fn.sion  and  Ebullition,  chictlv  in  relation  to  pressure, 
by  Carl  Barus.     1893.     8°.     57  pp.     9  pi.     Price  10  cents. 

104.  Glaciation  of  the  Yellowstone  Valley  north  of  the  Park,  by  Walter  Harvey  Weed.    1893.    8'^. 
41  pp.     4  pi.     Price  5  cents. 

105.  The  Laramie  and  the  overlying  Livingstone  Formation  iu  Montana,  by  Walter  Harvey 
Weed,  with  Report  on  Floia,  by  Frank  Hall  Knowlton.     1893.    8^    68  pp.     0  ]>1.     Price  10  cents. 

106.  The  Colorado  Formation  and  its  Invertebrate  Fauna,  by  T.  \V.  Stanton.     1893.     8'^.    288 
pp.     45  pi.     Price  20  cents. 

107.  The  Traj)  Dikes  of  Lake  Champlain  Valley  and  the  Eastern  Adirondacks,  by  James  Furuuin 
Kemp. 

108.  A  Geological  Reconnoissauce  in  Central  Washington,  by  Israel  Cook  Russell.     1893.     8". 
108  pp.     12  pi.     Frice  15  cents. 

109.  The  Eruptive  and  Sedimentary  Rocks  on  Pigeon  Point,  Minnesota,  and  their  contact  phe- 
nomena, by  AVilliam  Shirley  Bayley.     1893.     8^.     121  pp.     16  pi.     Price  15  cents. 

110.  The  Paleozoic  Section  in  the  vicinitv  of  Three  Forks.  Moutana,  bv  Albert  Charles  Peale. 
1893.     8°.    56  pp.     6  pi.     Price  10  cents. 

In  press : 

102.  A  Catalogue  and  Bibliography  of  North  American  Mesozoic  Invertebrata,  by  C.  B.  Boyle. 

111.  Geology  of  the  Big  Stone  Gap  Coal  Fields  of  VirgL.ia  and  Kentucky,  by  Marius  R.  Camii- 
bell. 

112.  Earthquakes  itf  California  in  1892.  by  Charles  D.  Perriue. 

In  preparation ; 

—  Correlation  papers — Pleistocene,  by  T.  C.  Chambeiiin. 

—  The  Moraines  of  the  Missouri  Coteau  and  their  attendant  deposits,  by  James  Edward  Todd. 

—  On  the  Structure  of  the  Ridge  between  the  Tacimic  and  the  Gieen  Mountain  Ranges  in  Ver- 
mont; and  On  the  Structure  of  Monument  Mouutaiu  in  Great  Barriugtoii,  Mass.,  by  T.  Nelson  Dale. 

—  A  Bibliography  of  Paleobotany,  by  David  White. 


VI  ADVERTISEMENT. 

STATISTICAL  PAPERS. 

Mineral  Resource.^  of  thp  UuiteaState.^  [1882],  by  Albert  Williams, , jr.     1883.     8'^.     xvii,813pp. 

Price  .50  eeuts. 

Mineral  Resources  ol' the  Uuiteil  State.s  1883  iiud  1884,  liy  AUiert,  Williams,  jr.     1885.     8^.     xiv, 

1011)  pp.     Price  60  cents.  .     .  ,  ,„     ,      , 

Mineral  R-sources  of  the  United  States,  1885.     Divi.'iion  of  Mining-  Statistics  and   Technology. 

1886.     8°.     vii,  .576  pp.     Price  40  cents.  '  ...,,, 

Mineral  Resources  of  the  United  States,  1886,  by  Havid  T.  Day.     188(.     8'^.     viii,813pp.     Price 

Mineral  Resources  of  the  United  States,  1887,  by  David  T.  Day.     1888.     8".     vii,  832  pp.     Price 

Mineral  Resources  of  the  United  States,  1888,  by  liavid  T.  Day.     18R0.     8".     vii,  6.52  pp.     Price 

^^"ilineral  Resources  of  the  United  States,  1889  and  1890,  by  David  T.  Day.     1892.     8°.     viii,  671  pp. 

""^""Mineral'Resourceaof  the  United  States,  1891,  by  David  T.  Day.     1893.     S^'.     vii,  630  pp.    Price 
50  cents. 

The  money  received  from  the  sale  of  these  publications  is  deposited  in  the  Treasury,  and  the 
Secretary  of  tha't  Departmeut  declines  tosreceive  bank  checks,  drafts,  or  postage-stamps;  all  remit- 
tances, therefore,  inn.st  be  by  POSTAL  XOTK  or  money  order,  made  payable  to  the  Chief  Clerk  of  the 
IT.  S.  Geological  Survey,  or  m  curhkncy  for  the  exact  amount.  Correspondence  relating  to  the  pub- 
lications of'the  Survey  should  be  addressed 

To  THE  Director  of  thv; 

United  States  (Jeological  Survey, 

Washington,  D.  C. 
Washington,  D.  C,  Uclohei;  1893. 


DEPARTMENT    OF   THE    INTERIOR 


MONOGRAPHS 


United  States  Geological  Survey 


VOLUME   XXII 


WASHINGTON 

GOVERNMENT    PRINTING    OFFICE 

1893 


1/1 12A 


UNITED    STATES    GEOLOGICAL    SURVEY 

J.    W.  POWELL,  DIRECTOR 


A    MANUAL 


TOPOGRAPHIC    METHODS 


HENRY    GANNEXT 

CHIEF     TOFOGRA.PHER 


WASHINGTON 

GOVERNMENT    PRINTING     OFFICE 
1893 


CONTENTS. 


Page. 

Letter  of  transmittal 

Chapter  I. 
Introduction 

Surveys  under  the  U.  S.  Government : - 

Exploration  of  the  Fortieth  Parallel 2 

Geologic  and  Geographic  Survey  of  the  Territories  2 

Geologic  and  Geographic  Survey  of  the  Rocky  Mountain  region -  -  3 

Northern  Transcontineutal  Survey 3 

Coast  and  Geodetic  Survey 

Engineer  Corps,  U.  S.  Army * 

General  Land  Office  Surveys * 

Surveys  under  State  governments - 

Massachusetts 

New  York ^ 

5 

New  Jersey 

5 

Pennsylvania 

5 
Railroad  and  other  surveys 

Plan  of  the  map  of  the  United  States 

Scale 

Scales  of  topographic  maps  of  European  nations 9 

9 

Contour  interval ■ 

9 
Features  represented 

Size  of  sheets - 

Geometric  control - 

Its  accuracy 

Its  amount 

Its  distribution 

14 
Sketching 

Chapter  II. 

15 
Classification  of  work - 

Astronomic  determinations  of  position 

17 
Definitions 

Astronomical  transit  and  zenith  telescope 

19 
Chronograph   

20 
Field  work 


YI  CONTENTS. 

Astronomic  determinations  of  position — Continued.  Page. 

Observations  for  Latitude 21 

Reduction  of  observations  for  latitude 23 

Measurement  of  a  division  of  the  head  of  the  micrometer  screw 23 

Measurement  of  a  level  division 26 

Computation  of  apparent  declination  of  stars 27 

Computation  of  Latitude 28 

Observations  for  time  28 

Eeduction  of  time  observations 29 

Correction  for  error  of  level 29 

Correction  for  inequality  of  pivots , 30 

Correction  for  error  of  collimation 30 

Correction  for  deviation  in  azimuth 30 

Correction  for  diurnal  aberration 31 

Comparison  of  time 34 

Observ.ations  for  azimuth 36 

Eeduction  of  observations  for  azimuth  38 

Chapter  III. 

Horizontal  location 41 

Party  organization 41 

Base  line  measurement 42 

Eeduction  of  base  line  measurement 45 

Eeduction  to  standard 45 

Correction  for  inclination  46 

Correction  for  temperature 46 

Reduction  to  sea  level 46 

Primary  triangulation 48 

Selection  of  stations 49 

Signals ; 50 

Heliotropes 52 

Theodolites  for  triangulation 54 

Instructions  for  the  measurement  of  horizontal  angles 55 

Organization  of  parties  and  prosecution  of  work 63 

Eeduction  of  primary  triangulation 65 

Reduction  to  center 65 

Spherical  excess 65 

Station  adjustment 66 

Figure  adjustment 68 

Computation  of  distances 72 

Computation  of  geodetic  coordinates 72 

Traverse  lines  for  primary  control , .  75 

Primary  elevations 77 

Chapter  IV. 

Secondary  triangulation 79 

The  plane  table 79 

The  alidade 82 

Measurement  of  altitudes 84 


CONTENTS.  VII 


Traverse  work °^ 

Traverse  plane  table 86 

Measurements  of  altitudes  in  c  mnection  with  traverse  work '. 89 

The  aneroid - ---  9^ 

Organization  of  parties  and  distribution  of  work 91 


Stadia  measurements. 


92 


The  Cistern  barometer 9"* 

Use  in  field ^5 

Reduction  of  barometric  observations  .  - . : - 98 

Utilization  of  the  work  of  the  public  land  surveys 101 

Description  of  work 102 

Chapter  V. 

Sketching l*'^ 

Origin  of  topographic  features _ - 108 

UpUffc 1"^^ 

Deposition  from  volcanic  action HO 

Aqueous  agencies 

Erosion ^^^ 

Weathering 1' ^ 

Transportation  and  corrasion HI 

Profiles  of  streams  and  of  the  terrane 112 

Relations  between  stream  and  terrane  corrasion 113 

Origin  of  canyons  in  plateau  region H'i 

Origin  of  detrital  vaUeys 115 


115 
116 


Sinks : 

Piracy ---- 

Origin  of  canyons  in  mountain  ranges 11° 

Origin  of  water  and  wind  gaps 116 

Junctions  of  streams 11'^ 

Effect  of  structure  on  topographic  forms 117 

Erosion  of  horizontal  beds  of  rock 118 

Erosion  of  inclined  beds  of  rock 1-0 

Age  of  topographic  features l-O 

Conception  of  base  level - l-l 

121 
Deposition  from  water : 

121 
River  ridges 

12^ 
Alluvial  fans " 

„      „„„„  122 


122 
Glacial  deposition 

123 
Drunvlins 

123 
Pitted  plains 

„  123 

Osars 

,,      .  123 

Moraines 

123 
Glacial  erosion 

124 
Amphitheaters ' 

Deposition  from  the  atmosphere 


VIII 


CONTENTS. 


Scale  of  ficldwork 

Reports 

Inspection 


Chapter  VI. 


Office  trork 

Form  of  original  sheets 

Construction  of  projections . 

Colors  and  conventions 

Titles  and  legends 


Pago. 
125 
125 
127 

128 
128 
129 
130 
130 


TABLES 


Page. 
Table   I.  For  computing  the  difference  in  the  heights  of  two  places  from  barometric 

observations 1^1 

II.  Correction  for  the  difference   of  temperature  of  the  barometers  at  the  two 

stations ' ■'•'* 

III.  Correction  for  the  difference  of  gravity  in  various  latitudes 134 

IV.  Correction  for  decrease  of  gravity  on  a  vertical 135 

V.  Correction  for  the  height  of  the  lower  station 135 

VI.  Differences  of  altitude  from  angular  measurements  for  low  angles  and  short 

distances '^^" 

VII.  Differences  of  altitude  from  angular  measurements  for  unit  distance  and  high 

angles 1°-^ 

VIII.  Corrections  for  curvatvire  and  refraction 153 

IX .  Differences  of  altitude  from  angular  measurements  applicable  to  scale  1 :  45000 .  154 

X.  Differences  of  altitude  from  angular  measurements  applicable  to  scale  1 :  30000.  156 

XI.  Differences  of  altitude  from  telemeter  measurements 158 

XII.  For  converting  wheel  revolutions  into  decimals  of  a  mile 162 

XIII.  Constants 163 

XIV.  Conversion  table— metres  into  yards 163 

XV.  yards  into  metres ■ 164 

XVI.  inches  into  metres  and  metres  into  inches 164 

XVII.  metres  into  statute  and  nautical  miles 164 

XVIII.  statute  and  nautical  miles  into  metres 164 

XIX.  Coordinates  for  projection  of  maps  of  large  areas 165 

XX.  Coordinates  for  projection  of  maps,  scale  1 :  250000 175 

XXI.  Coordinates  for  projection  of  maps,  scale  1 :  125000 177 

XXII.  Coordinates  for  projection  of  maps,  scale  1 :  62500 180 

XXIII.  Coordinates  for  proj  ection  of  maps,  scale  1 :  45000 185 

XXIV.  Areas  of  quadrilaterals  on  the  earth's  surface,  one  degree  in  latitude  and  in  lon- 

gitude    186 

XXV.  Areas  of  quadrilaterals  ou  the  earth's  surface,  30  minutes  of  latitude  and  longi- 

tude    18'^ 

XXVI.  Areas  of  quadrilaterals  on  the  earth's  surface,  15  minutes  of  latitude  and  longi- 
tude    188 

XXVII.  Factors  for  the  geodetic  computation  of  latitudes,  longitudes,  and  azimuths. . .  190 

XXVIII.  Factors  for  reduction  of  transit  observations 217 

XXIX.  For  reducing  observations  for  latitude  by  Talcott's  method 224 


:  TABLES. 

Page. 
Tablk     XXX.  For  facilitating  tiie  reduction  of  observations  on  close  circum-polar  stars  made 

in  determining  the  value  of  a  revolution  of  the  micrometer 226 

XXXI.  For  converting  sidereal  time  into  mean  time 227 

XXXII.  For  converting  mean  time  into  sidereal  time 228 

XXXIII.  For  converting  parts  of  the  equator  in  arc  into  sidereal  time 229 

XXXIV.  For  converting  sidereal  time  into  parts  of  the  equator  in  arc 230 

XXXV.  Logarithms  of  numbers 231 

XXXVI    Logarithms  of  circular  functions 254 


ILLUSTRATIONS 


Page. 
14 


50 
54 
80 
86 
112 
113 
114 


Plate  I.  Map  of  surveyed  areas.     Folded  in  pocket 

II.  Diagram  of  control 

III.  Baldwin  base-measuring  device ** 

IV.  Signal 

V.  Eight-inoli  theodolite  and  tripod 

VI.  Johnson  plane-table— general  view 

VII.  Traverse  plane-table 

VIII.  Types  of  topography,  Great  plains 

IX.  Types  of  topography,  Atlantic  plain 

X.  Types  of  topography,  Cumberland  plateau 

XI.  Types  of  topography.  Canyons  in  homogeneous  rocks US 

XII.  Types  of  topography,  Canyons  in  rocks  not  homogeneous 116 

XIII.  Types  of  topography.  Grand  canyon  of  Colorado  river -  -  -       117 

XIV.  Types  of  topography.  Water  gaps,  Pennsylvania 118 

XV.  Types  of  topography,  Mississippi  river  ridge 121 

XVI.  Types  of  topography,  Drumlins 1^2 

XVII.  Types  of  topography,  Moraines .- :       1-"^ 

XVIII.  Types  of  topography,  Cirques 1^* 

Figure  1.  Astronomical  transit  and  zenith  telescope :  -  •  1° 

2.  Chronograph 

3.  Switchboard ^ 

4.  Signal  and  instrument  support 

5.  Heliotrope,  Coast  Survey = ^-' 

6.  Heliotrope,  Steiuheil ^^ 

7.  Eight-inch  theodolite— detail ^^ 

8.  Johnson  plane-table  tripod  head— section 81 

87 

9.  Douglas  odometer - 

10.  Small  telescopic  aUdade - 

11.  Aneroid - 

12.  Aneroid -^ ^'^ 

i-in 

13.  Cross  sections  of  canyons 

14.  Cross  sections  in  inclined  beds I-'" 


LETTER    OF   TRANSMITTAL 


Department  of  the  Interior, 

U.  S.  Geological  Survey, 

Geographic  Branch, 
Washington,  D.  C,  May  17,  1892. 

Sir:  I  have  the  honor  to  submit  herewith  for  pubhcation  a  manual  of 
the  topoga-aphic  methods  in  use  by  the  Geological  Survey,  accompanied  by 
a  collection  of  constants  and  tables  used  in  the  reduction  of  astronomical 
observations  for  position,  of  triang-alation,  of  height  measurements,  and 
other  operations  connected  with  the  making  of  topographic  maps.  It  must 
be  understood  that  the  methods  are  not  fixed,  but  are  subject  to  change  and 
development,  and  that  this  manual  describes  the  stage  of  development 
reached  at  present. 

In  the  preparation  of  this  work  I  have  to  acknowledge  the  aid  of  many 
of  my  associates,  notably  Mr.  H.  M.  Wilson  and  Mr.  S.  S.  Gannett.  To 
Mr.  R.  S.  Woodward,  now  connected  with  the  U.  S.  Coast  and  Geodetic 
Survey,  I  am  indebted  for  the  "  Instructions  for  the  Measurement  of  Hori- 
zontal Angles  "  in  Chapter  iii.  These  instructions,  which  were  di-awn  up 
by  Mr.  Woodward  several  years  ago  for  the  guidance  of  field  parties  en- 
gaged in  primary  triangulation,  have  resulted  in  a  great  increase  in  accuracy 
and  considerable  economy  of  time  and  labor.  To  Messrs.  G.  K.  Gilbert 
and  W.  J.  McGee  I  am  indebted  for  their  kindly  criticism,  especially  con- 
cerning the  chapter  upon  the  "  Origin  of  Topographic  Features." 


XIV  LETTEE  OF  TKANSMITTAL.  ^ 

'.I 
Some  of  the  tables  liave  been  prepared  in  this  office ;  others  have  been  ^ 

compiled  from  various  sources,  notably  from  appendices  to  reports  of  the  ; 

U.  S.  Coast  and  Geodetic  Survey  and  "Lee's  Tables  and  Formulae." 

Very  respectfully, 

Henry  Gannett, 

Chief  To;pograplier. 

Hon.  J.  W.  Powell, 

Director  U.  S.  Geological  Survey. 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


By  Henry  Gannett. 


CHAPTER   I. 

INTRODUCTION. 

The  object  of  this  manual  is  to  present  a  description  of  the  topographic 
work,  instruments,  and  methods  used  by  the  U.  S.  Greological  Survey, 
primarily  for  the  information  of  the  men  engaged  upon  this  work.  It  is 
not  intended  as  an  elementary  treatise  upon  surveying,  as  it  presupposes  a 
knowledge  of  the  application  of  mathematics  to  surveying  equivalent  to 
that  to  be  obtained  in  our  professional  schools.  Neither  is  it  intended  as  a 
general  treatise  on  topographic  work,  although  it  Tnay,  to  a  certain  extent, 
supply  the  existing  need  of  such  a  work. 

The  Geological  Survey  is  engaged  in  making  a  topographic  map  of  the 
United  States.  Excepting  for  certain  areas,  lying  mainly  in  the  far  West, 
there  existed,  prior  to  the  inception  of  this  work,  no  maps  upon  a  sufficiently 
large  scale  and  in  suitable  form  for  the  use  of  the  geologist.  While  the 
primary  object  of  the  map  is  to  meet  the  needs  of  the  geologists  of  the 
Survey,  it  has  been  thought  economical  to  adjust  the  plans  so  that  the  result- 
ing map  may  be  adequate  to  serve  all  needs  for  which  general  tojjographic 
maps  are  used. 

Certain  areas,  especially  in  the  far  West,  have  been  surveyed  and 
mapped  by  other  organizations,  notably  those  of  the  general  and  state  gov- 
ernments, upon  a  sufficiently  large  scale,  and  with  sufficient  accuracy  for 
the  use  of  the  Geological  Survey;  much  material  also  exists  in  the  form  of 
triangulation,  of  lines  of  levels,  and  of  other  partial  surveys  which  can  be 


2  A  MANUAL  OF  TOPOUEAPHIC  METHODS. 

put  to  use  aud  will  assist  to  a  greater  or  less  extent  iu  the  preparation  of 
the  map.  These  maps  and  other  material  have  been,  or  aiay  be,  adopted 
b}^  the  Geological  Survey.  Their  extent  is  represented  upon  the  accom- 
panying map,  PL  I,  as  fully  as  possible,  and  they  are  enumerated,  with  a 
brief  description,  as  follows: 

SURVEYS  UNDER  THE  UNITED  STATES  GOVERNMENT. 

The  Survey  of  the  Fortieth  Parallel,  from  1867  to  1872,  under  Mr.  ■ 
Clarence  King,  embraced  a  zone  of  country  105  miles  in  breadth,  extend- 
ing from  the  meridian  of  104°  to  that  of  120°  west  of  Greenwich,  and 
comprising  an  area  of  87,000  square  miles.  The  maps  were  made  upon  a 
scale  of  4  miles  to  an  inch,  with  contours  having  a  vertical  interval  of  300 
feet.  The  work  was  controlled  by  triangulation,  resting  primarily  upon  a 
base  line  measured  by  determining  astronomically  the  latitudes  of  two 
points,  and  the  azimuth  of  the  line  connecting  them ;  and,  secondarily,  upon 
a  base  line  extending  neai-ly  from  the  eastern  to  the  western  limits  of  the 
work,  the  coordinates  of  the  ends  of  which  were  determined  astronomically, 
the  latitude  by  zenith  telescope  and  the  longitude  by  telegraphic  time  com- 
parisons. Primary  triangulation  was  done  with  theodolites  reading  to  ten 
seconds.  Secondary  triangulation  and  location  were  executed  with  minute 
reading  instruments,  and  topography  was  sketched  and  afterwards  trans- 
fen-ed  to  the  platted  framework.  Heights  were  measured  by  barometer  and 
the  vertical  arc. 

The  Geological  and  Geographical  Survey  of  the  Territories,  under 
Dr.  F.  V.  Hayden,  between  1873  and  1878,  surveyed  areas  in  Colorado, 
New  Mexico,  Utah,  Wyoming,  Idaho,  in  all  about  100,000  square  miles. 
The  maps  were  published  \ipon  a  scale  of  4  miles  to  an  inch,  with  a  contour 
interval  of  200  feet.  The  base  lines  for  the  control  of  this  work  were 
measured  with  steel  tapes,  under  imiform  tension,  and  with  corrections  for 
temperature.  Triangulation  was  carried  on  with  ,8 -inch  theodolites  read- 
ing to  ten  seconds,  and  was  adjusted  by  a  graphic  method.  Secondary 
triangulation,  the  location  of  topographic  details,  and  the  measurement  of 
heights  were  effected  by  methods  quite  similar  to  those  employed  by  the 
Survey  of  the  Fortieth  Parallel. 


PEEVIOUS  MAPS.  3 

The  Survey  of  the  Kocky  Mountaiu  Region,  under  Maj.  J.  W.  Powell, 
embraced  an  area  of  about  60,000  square  miles,  covering  parts  of  Wyoming, 
Utah,  and  Arizona.  This  work  was  done  between  1869  and  1877.  The 
maps  Avere  drawn  upon  a  scale  of  4  miles  to  an  inch,  with  contour  intervals 
of  250  feet.  The  work  was  controlled  by  triangulation  from  base  lines 
measured  with  wooden  rods.  It  was  carried  on  with  a  theodolite  having  a 
10-inch  circle  reading  by  vernier  to  ten  seconds,  and  was  adjusted  by  the 
method  of  least  squares.  Secondary  triangulation  was  done  with  minute 
reading  instruments,  and  minor  locations,  together  with  topographic  details, 
were  obtained  by  the  use  of  the  plane  table.  Heights  were  measured  by 
the  barometer,  supplemented  by  the  vertical  circle. 

The  Northern  Transcontinental  Survey,  an  organization  instituted  by 
the  Northern  Pacific  railroad  company  for  the  survey  and  examination  of 
its  lands,  mapped,  during  the  years  1882  and  1883,  areas  in  Montana,  Idaho, 
and  Washington,  aggregating  about  43,000  square  miles.  These  maps  were 
intended  for  publication  upon  a  scale  of  4  miles  to  an  inch,  with  contours 
haAang'  vertical  intervals  of  200  feet.  The  work  was  based  upon  triangu- 
lation, which  was  executed  mainly  with  a  theodolite  having  a  circle  8 
inches  in  diameter  reading  by  vernier  to  ten  seconds.  The  triangulation 
was  adjusted  graphically.  The  topographic  methods  were  quite  similar  to 
those  of  the  Hayden  Survey. 

The  U.  S.  Coast  and  Geodetic  Survey  has  covered  the  greater  part  of 
the  Atlantic,  Gulf,  and  Pacific  coasts  with  triangulation,  and  with  a  narrow 
strip  of  topographic  work.  This  strip  is  of  variable  width,  depending 
largely  upon  the  configuration  of  the  coast,  being,  as  a  rule,  narrow  where 
the  coast  is  simple,  and  '1>i-oad  where  it  is  complex.  Altogether  an  area  of 
nearly  40,000  square  miles  has  been  surveyed,  the  original  sheets  being 
upon  a  scale  of  1:10000  or  1:20000,  the  contours  having  vertical  intervals 
of  20  feet.  Most  of  this  Avork  is  directly  available  as  finished  Avork.  Upon 
some  of  it,  howcA^er,*  the  contours,  owing  to  the  great  age  of  the  original 
maps,  have  been  obliterated,  and  it  becomes  necessary  to  wesurvey  this  ele- 
ment. In  addition  to  its  coast  work,  the  geodetic  Avork  of  this  orgaitization 
has  been  extended  into  the  interior  in  A-arious  directions,  especially  in  New 
England,  and  along  the  eastern  border  of  the  Appalachian  IMountiiin  system, 


4  A  MAIs^UAL  OF  TOPOGEAPHIC  METHODS. 

througli  the  states  of  New  York,  New  Jersey,  Pennsyh^ania,  Maryland, 
Yirgiuia,  West  Virginia,  North  Carohua,  Tennessee,  Georgia,  and- Alabama. 
The  work  of  connecting  the  Atlantic  and  Pacific  coasts  has  been  carried 
far  toward  completion,  a  belt  having  been  extended  westward  from  the 
head  of  Chesajjeake  Bay  into  centi-al  Kansas.  A  base  has  been  measured 
near  Colorado  Springs,  Colorado,  and  work  has  been  extended  thence  east- 
ward to  the  east  boundary  of  the  state,  while  from  the  Pacific  coast  triangu- 
latiou  has  been  brought  eastward  across  California,  Nevada  and  Utah. 

Ill  assisting  the  state  sui-veys,  the  Coast  and  Geodetic  Survey  has, 
moreover,  done  a  considerable  amount  of  triangulation  in  the  states  of  Mas- 
sachusetts, New  York,  New  Jersey,  Pennsylvania,  Kentucky,  Tennessee, 
and  Wisconsin. 

The  United  States  Lake  Siu-vey  has  mapped  the  shores  of  the  Great 
lakes,  caiTying  triangulation  around  them,  and  connecting  the  head  of  Lake 
Michigan  with  the  foot  of  Lake  Erie.  A  belt  of  triangulation  has  also  been 
can-ied  from  the  neighborhood  of  Vincennes,  Indiana,  northward  along  the 
eastern  border  of  Illinois,  connecting  with  the  triangulation  on  the  shore  of 
Lake  Michigan. 

The  Engineer  Corps,  U.  S.  Army,  has  completed  a  number  of  small 
pieces  of  topogi-aphic  work  in  different  parts  of  the  country,  and  is  now 
engaged  in  mapping  the  com-se  of  the  Mississippi  and  Missouri  rivers,  con- 
trolling the  work  by  geodetic  methods. 

The  surveys  of  the  General  Land  Oflice  have  extended  over  an  area 
of  about  a  million  and  a  half  square  miles,  and  plats  have  been  prepared 
representing  the  drainage  of  this  entire  area.  The  quality  of  this  work  is 
of  varying  degrees  of  excellence,  but  from  its  inception  in  the  early  part 
of  the  centurr  its  quality  has  improved  greatly.  Most  of  this  Avork  can  be 
utilized  either  directly  or  indirectly  by  methods  to  be  detailed  hereafter. 

SURVEYS  UNDER  STATE  GOVERNMENTS. 

Massadms^ts. — Between  1830  and  1842,  the  state  of  Massachusetts 
carried  on  what  was  for  the  time  an  elaborate  system  of  triangulation, 
known  as  the  Borden  Survey.  By  this  organization  numerous  points,  in 
the  aggregate  several  hundred,  were  determined  within  the  limits  of  the 


PEEVIOUS  MAPS.  5 

state.  Subsequently,  many  of  these  points  were  redetermined  by  the 
Coast  and  Greodetic  Survey,  by  more  elaborate  methods,  thus  furnishing 
what  served  substantially  as  a  primary  system  of  triangulation  within  which 
and  to  which  the  Borden  work  has  been  adjusted.  As  thus  adjusted,  the 
Borden  locations  are  sufficiently  accurate  for  the  ordinary  needs  of  map 
work  upon  the  scale  of  one  mile  to  an  inch. 

New  York — For  several  years,  terminating  in  1885,  the  state  of  New 
York  supported  a  survey  which  was  devoted  to  the  geodetic  location  of 
points  within  its  area.  The  work  was  of  a  high  grade,  comparing  favora- 
bly with  that  of  the  Coast  and  Greodetic  and  Lake  Surveys. 

For  many  years  also,  the  same  state  supported  what  was  known  as  the 
Adirondack  Survey,  which  was  engaged  mainly  in  a  triangulation  of  the 
Adirondack  region.     Of  this  work  few  results  have  been  published. 

New  Jersey. — In  the  year  1875,  the  state  of  New  Jersey  instituted  a 
topographic  survey  of  its  area.  The  plan  of  the  work  contemplated  a  map 
upon  a  publication  scale  of  one  mile  to  an  inch,  with  contours  at  vertical 
intervals  ranging  from  5  to  20  feet.  Control  of  the  work  was  furnished  in 
part  by  the  triang-ulation  of  the  Coast  and  Geodetic  Survey  and  in  part  by 
triangulation  of  its  own.  In  July,  1884,  the  completion  of  that  state  was 
undertaken  by  the  U.  S.  Greological  Survey,  by  which  organization  it  was 
pushed  forward  to  a  conclusion  in  1887. 

Pennsylvania. — In  Pennsylvania  considerable  topographic  work  has 
been  done  by  the  State  Greological  Survey.  This  woi'k  is  of  a  local  char- 
acter and  confined  to  small  areas,  which  have  been  mapped  upon  large 
scales,  and  the  ag'g'regate  area  is  not  large.  It  was  carried  on  by  traverse 
by  the  use  of  stadia  and  level. 

RAILROAD  AND   OTHER  SURVEYS. 

Besides  the  material  above  enumerated,  there  exist  in  various  parts  of 
the  country  maps  in  great  number  and  of  varying  quality.  They  consist  of 
town  and  county  maps,  mainly  made  by  traversing  roads  with  odometer 
and  compass,  of  railroad  lines,  executed  in  the  ordinary  manner  by  transit 
and  chain,  the  surveys  of  the  boundaries  of  the  states  and  territories,  etc. 
Some  of  this  material  may  prove  of  service. 


6  A  MANUAL  OP  TOPOGRAPHIC  METHODS. 

In  additiou  to  the  material  enumerated  above,  numerous  astronomic 
determinations  of  position  have  been  made  by  governmental  organizations 
and  by  private  parties.  These  positions,  scattered  over  the  interioi",  will,  as 
far  as  they  go,  relieve  the  Greological  Survey  from  carrying  on  this  expen- 
sive work. 

In  additiou  to  all  this  material,  the  railroads  of  the  country  furnish,  in 
their  profiles,  a  vast  bod}^  of  measurements  of  height.  These  differ  greatly 
in  value,  those  of  certain  railroads,  and  generally  those  of  the  great  systems, 
being  of  a  high  degree  of  accuracy,  while  others  are  worthless.  The  errors 
in  these  profiles  are  seldom  in  the  leveling  itself,  but  are  due  to  the  fact 
that  a  road  is  leveled  in  sections,  the  profile  of  each  section  being  based 
upon  an  arbiti'ary  datum  point.  Mistakes  often  occur  in  joining  the  profiles 
of  the  several  sections,  and  in  correcting  them  for  diff'erences  in  their  datum 
points. 

PLAN   OF  THE  MAP  OF  THE  UNITED  STATES. 

The  field  upon  which  the  Geological  Survey  is  at  work  is  diversified. 
It  comprises  broad  plains,  some  of  which  are  densely  covered  with  forests, 
while  upon  others  trees  are  entirely  absent.  It  contains  high  and  rugged 
mountains,  plateaus,  and  low,  rolling  hills.  In  some  regions  its  topographic 
forms  are  upon  a  grand  scale,  while  in  others  the  entire  surface  is  made  up 
of  an  infinity  of  minute  detail.  Some  parts  of  the  country  are  densely 
populated,  as  much  so  as  almost  any  region  upon  the  surface  of  the  globe, 
while  great  areas  in  other  parts  of  the  country  are  almost  without  settle- 
ment. Greologically,  portions  of  the  country  are  extremely  complex,  re- 
quiring, for  the  elucidation  of  geologic  problems,  maps  in  great  detail,  while 
other  areas  are  simple  in  the  extreme. 

It  is  ob^'ious  that  from  this  diversity  of  conditions,  both  natural  and 
material,  maps  of  different  areas  should  differ  in  scale,  and  that  with  the 
difference  in  natural  conditions  and  the  difference  in  scale  there  must  come 
differences  in  the  methods  of  work  employed.  The  system  which  is  found 
to  work  to  advantage  in  the  high  mountain  regions  of  the  west  is  more  or 
less  inapplicable  to  the  low  forested  plains  of  the  Mississippi  valley  and  the 
Atlantic  plain. 


PLAN  OP  THE  MAP, 


The  scales  which  have  finally  been  adopted  for  the  publication  of  the 
map  are  1:62500  or  very  nearly  1  mile  to  an  inch,  and  1:125000,  or  very 
nearly  2  miles  to  an  inch. 

When  this  work  was  commenced  in  18H2,  three  different  scales  were 
used  for  different  parts  of  tlie  country,  depending  upon  the  degree  of  com- 
plexity of  the  topography  and  the  geological  phenomena,  upon  the  density 
of  population  and  the  importance  of  the  region  from  an  industrial  point  of 
view.  These  scales  were  1:62500,  1:125000,  and  1:250000.  The  luaps  as 
fast  as  produced  have  found  extended  use,  not  only  among  geologists,  but 
in  all  sorts  of  industrial  enterprises  with  which  the  surface  of  the  ground  is 
concerned,  and  have  abeady  become  well  nigh  indispensable  in  the  pro- 
jection of  railroads,  water  works,  drainage  works,  systems  of  irrigation,  and 
other  similar  industrial  enterprises.  Their  extended  use  has  developed  a 
requirement  for  better  maps;  i.  e.,  maps  upon  a  larger  scale  and  in  greater 
detail.  At  one  stage  of  its  development  this  requirement  was  met  by  dis- 
continuing all  mapping  upon  the  scale  1:250000,  which  it  was  recognized  at 
that  time  was  inadequate  to  the  requirements.  Since  then  the  standard  of 
the  requirements  has  continued  to  rise  and,  consequently,  the  plan  of  the 
map  has  been  enlarged  by  the  extension  of  the  areas  mapped  upon  the  scale 
of  1:62500,  and  a  corresponding  reduction  of  the  areas  to  be  mapped  upon 
the  scale  of  1:125000.  Meantime,  however,  large  areas  have  been  mapped 
upon  the  discarded  scale,  and  the  maps  have  been  published  and  widely 
distributed.  Such  areas  will  be  remapped  for  the  larger  scales  only  as 
special  needs  may  arise. 

The  considerations  which  have  determined  the  selection  of  the  above 
scales  are  as  follow§:  They  are  believed  to  be  sufficiently  large  to  represent 
with  faithfulness  all  the  details  required  to  picture  the  country  and  show  the 
proper  relations  of  its  features,  and  to  make  the  map  of  the  greatest  pos- 
sible service  for  industrial  and  scientific  uses  consistent  with  other  require- 
ments to  be  mentioned  hereafter.  These  scales  are  sufficiently  large  to 
present  the  details  of  nearly  all  geological  phenomena.  The  map  represents 
the  country  in  sufficient  detail  to  admit  of  the  selection  upon  it  of  general 
routes  for  railroads  and  other  jiublic  Avorks  and  to  show  the  location  of 


8  A  MANlTxiL  OF  TOPOGRAPHIC  METHODS. 

boundary  lines  in  such  way  that  their  position  may  be  recognized  upon  the 
ground.  On  the  other  hand,  the  scales  are  not  so  large  as  to  prevent  the 
representation  upon  a  single  sheet  of  a  considerable  area,  so  that  the  rela- 
tions between  different  regions  can  be  seen  at  a  glance. 

A  map  on  a  larger  scale  than  this  would  require  a  greater  time  for  its 
completion  and  a  greater  expense,  and  when  one  considers  the  fact  that  the 
map  upon  these  scales  of  the  entire  United  States,  even  excluding  Alaska, 
will,  at  best,  cost  in  the  neighborhood  of  twenty  million  dollars  and  at  the 
present  rate  of  progress  require  fifty  years  for  its  completion,  one  scarcely 
feels  inclined  to  increase  the  labor  and  expense  without  an  excellent 
reason  for  so  doing.  There  is  yet  another  objection  to  increasing  the  scale 
beyond  that  absolutely  necessary.  To  be  of  value,  such  a  map  must  undergo 
revision  at  frequent  intervals,  in  order  to  incorporate  any  changes  in  culture 
and  possibly  in  natural  features  due  to  natm-al  or  artificial  agencies.  The 
larger  the  scale  the  more  frequently  such  revision  should  be  made,  and 
hence  the  labor  and  expense  of  keeping  a  map  up  to  date  would  be  greatly 
increased. 

In  this  matter  the  experience  of  the  ciAdlized  nations  of  Europe,  all  of 
which  have  prepared  topographic  maps  of  more  or  less  of  their  areas,  while 
certain  of  them  have  mapped  their  entire  areas  several  times,  is  of  great 
service  and  points  immistakably  in  the  direction  of  the  adopted  scales.  The 
history  of  these  nations  in  this  matter  presents  a  singular  degree  of  uni- 
formity. Their  first  maps  were  upon  large  scales,  and  upon  them  they 
attempted  to  represent  alh details  of  natural  and  artificial  topography,  even 
property  lines,  so  that  one  set  of  maps  would  answer  for  all  purposes.  Ex- 
perience of  the  difficulty  and  expense  of  keeping  up  such  maps  (without 
coimting  then-  original  cost)  has  taught  them  that  economy  consists  in  the 
production  of,  not  a  single  map,  but  a  series  of  maps,  each  designed  to  serve 
a  special  purpose ;  one  as  a  cadastral  map,  another  as  a  military  map,  and 
another,  and  this  the  most  important,  as  a  general  topographic  map.  It 
further  taught  that  this  topographic  map  shoukl  be  on  a  comparatively  small 
scale,  and  accordingly,  as  a  rule,  the  maps  of  foreign  countries  are  upon 
scales  approximating  one  mile  to  an  inch,  a  scale  which  is  sufficient  to  show 
all  topographic  details  of  a  general  character,  and  serves  all  ordinary  pur- 


PLAN  OP  THE  MAP.  9 

poses.     The  following  table  presents  the  scales  of  the  general  topographic 
maps  of  various  European  countries: 

Scales  of  lopographk  maps  of  European  nations. 

India 1:63360 

Great  Britain  and  Ireland 1 :  63360 

Germany 1 :  100000 

Austria-Hungary 1 :  75000 

France 1 :  80000 

Q    ■+      ,      1  S   1:25000 

Switzerland < 

I   1:50000 

Holland : 1 :  25000 

Spain 1 :  50000 

Italy 1:100000 

Swedea 1 :  100000 

Russia 1 :  126000 

1:20000 


\  1-A 


Belgium  , 

: 40000 

Denmark 1 :  40000 

Norway 1:100000 

Portugal 1 :  100000 

CONTOUE    INTERVAL. 

The  relief  of  the  earth's  surface  is  now  represented  upon  maps  almost 
entirely  by  contour  lines  or  lines  of  equal  elevation.  Until  a  comparatively 
recent  date  this  element,  secondary  in  importance  only  to  the  horizontal 
element,  or  the  plan,  has  been  ignored. 

The  contour  intervals  which  have  been  adopted  for  the  map  of  the 
United  States  are  as  follows: 

For  the  scale  of  1 :  62500,  the  intervals  range  from  5  to  50  feet;  for  the 
scale  of  1 :  125000, 10  to  100  feet,  and,  for  the  scale  of  1 :250000,  the  interval 
is  200  or  250  feet. 

FEATURES   REPRESENTED. 

In  this  matter,  the  experience  of  European  nations  tends  in  the  direc- 
tion of  reducing  the  number  of  features  which  should  be  placed  upon  the 
map.  It  has  been  decided,  in  the  preparation  of  the  map  of  the  United 
States,  to  go  even  beyond  the  present  practice  of  European  nations  in  this 
regard  and  to  limit  the  map  to  the  representation  of  all  natural  features 


10  A  MANUAL  OF  TOPOGEAPHIO  METHODS. 

wliicli  are  of  sufficient  maguitude  to  warrant  representation  upon  the  scale, 
and  to  confine  tlie  cultural  features,  that  is,  the  artificial  ones,  to  those  which 
are  of  general  or  public  importance,  leaving  out  those  which  are  jDrivate  in 
their  nature.  Under  this  definition  the  map  will  represent  cities,  towns,  and 
villages,  roads  and  railroads  and  other  means  of  communication  (with  the 
exception  of  private  roads),  bridges,  femes,  tunnels,  foixls,  canals  and 
acequias  and  boundaries  of  civil  divisions.  Fences,  property  lines,  private 
roads,  and  other  objects  of  a  kindred  nature  are  not  represented.  The 
reasons  for  excluding  priA^ate  culture  are  apparent.  They  are,  first,  because 
such  features  are  not  of  sufficient  general  interest  to  pay  the  cost  of  survey- 
ing or  representing  them;  second,  because  they  change  rapidly,  and,  in 
order  to  keep  the  maps  up  to  date,  would  require  constant  resurveys  and 
republication,  while  if  the  map  is  not  kept  constantly  up  to  date,  it  is  mis- 
leading, and,  third,  their  number  and  complexity  confuse  the  map  and 
render  its  more  important  features  less  intelligible. 

SIZE    OF   SHEETS. 

Atlas  sheets  are  designed  to  be  approximately  of  the  same  size,  17  5 
inches  in  length  by  from  12  to  15  in  breadth,  depending  upon  the  latitude, 
and  all  those  of  the  same  scale  cover  equal  areas,  expressed  in  units  of 
latitude  and  longitude,  that  is,  each  sheet  upon  the  4-mile  scale  covers 
one  degree  of  latitude  by  one  degree  of  longitude;  each  sheet  upon  the 
2-mile  scale,  30  minutes  of  latitude  and  longitude,  and  each  sheet  upon 
the  1-mile  scale,  15  minutes  of  latitude  and  longitude.  The  sheets  are 
thus  small  enough  to  be  conveniently  handled,  and,  if  bound,  form  an 
atlas  of  convenient  size.  From  the  fact  that  each  sheet  is  either  a  full 
degree  or  a  regular  integral  part  of  a  degree,  its  position  with  relation  to 
the  adjacent  sheets  and  to  the  area  of  the  country  is  easy  to  discovei'. 

GEOMETRIC    CONTROL. 

From  the  constructive  point  of  view,  a  map  is  a  sketch,  corrected  by 
locations.  The  work  of  making  locations  is  geometric,  that  of  sketching  is 
artistic.  This  definition  is  applicable  to  all  maps,  whatever  their  quality  or 
character.     However  numerous  the  locations  may  be,  they  form  no  part  of 


CONTEOL  OP  THE  MAPS.  H 

the  map  itself,  but  serve  only  to  correct  the  sketch,  while  the  sketch  sup- 
plies all  the  material  of  the  map.  The  correctness  of  the  map  depends 
upon  four  elements:  first,  the  accuracy  of  location;  second,  the  number  of 
locations  per  square  inch  of  the  map ;  third,  their  distribution ;  and,  fourth, 
the  quality  of  the  sketching.  It  is  in  connection  with  the  first  of  these 
elements  that  it  seems  desirable  to  define  what  constitutes  accuracy.  The 
greatest  accuracy  attainable  is  not  alwaj^s  desirable,  because  it  is  not 
economic.  The  highest  economy  is  in  properly  subordinating  means  to 
ends  and  it  is  not  economic  to  execute  triangulation  of  geodetic  refinement 
for  the  control  of  maps  upon  small  scales.  The  quality  of  the  work  should 
be  such  as  to  insure  against  errors  of  sufficient  magnitude  to  appear  upon 
the  scale  of  publication.  While  the  tendency  of  errors  in  triangulation  is 
to  balance  one  another,  still  they  are  liable  to  accumulate,  and  this  liability 
must  be  guarded  against  by  maintaining  a  somewhat  higher  degree  of 
accuracy  than  would  be  required  for  the  location  of  any  one  point.  It  is 
not  difficult  to  meet  this  first  condition  of  accuracy  of  the  maps  The 
maximum  allowable  error  of  location  may  be  set  at  one-hundredth  of  an 
inch  upon  the»scale  of  publication.  This  admits  of  an  error  upon  the  ground 
not  greater,  on  a  scale  of  1:62500,  than  50  feet. 

The  second  condition  of  correctness,  that  is,  the  number  of  locations 
necessary  for  the  proper  control  of  the  work,  is  not  easily  defined.  The 
requirements  difi'er  with  the  character  of  the  country.  A  region  of  great 
detail  and  of  abrupt  features  requires  more  control  than  one  of  great  uni- 
formity and  gentle  slopes  and  of  large  features,  so  that  no  general  rule  can 
be  laid  down.  Furthermore,  it  depends  upon  the  quality  of  the  sketching ; 
with  indifferent  sketching  a  greater  number  of  locations  is  required  in  order 
to  bring  the  map  up  to  the  requisite  quality. 

The  following  table  presents  a  summary  of  the  amount  of  control 
obtained  during  the  field  season  of  1891  in  the  diff"erent  fields  of  work  in 
this  survey.  It  is  presented  not  as  a  type  of  what  should  be,  but  to  show 
what  has  been  done  and  also  to  illustrate  the  wide  range  in  the  amount  of 
control  brought  about  by  the  differences  in  the  character  of  the  country' 
and  methods  of  work. 


12 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

Statistics  of  control ,  fu'W  neason  1S91. 


Area  surveyed,  square  miles 

Triangulation  statioas 

Kumber  of  square  inches  per  station 

Points  located  by  triimgulation 

Triangulation  stations  and  located  points  . 
Number  of  above  locations  per  square  inch 

Number  of  miles  traversed 

Incbes  traversed  per  square  inch  - . .' 

Number  of  traverse  stations 

Traverse  stations  per  square  inch 

Total  number  of  locations  per  square  inch. 

Traverse  stations  per  linear  mile 

Heights  measured  instrumentally 

Heights  measured  by  aneroid 

Total  number  of  measured  heights 

Heights  per  square  inch 


Northeast 
division, 

New 
Enu'laud, 
Vorlv 


and  Penn-      Atlantic 
sylvania.         Plain. 


Southeast 
division, 
Appalachi- 
an rosion 
an^ 


113,  600 
50. 1 
56.8 


48,  880 
56,  680 


Central  division. 


1,276 
4,034 
5,310 


3,450 
13, 100 
16,  556 


56.1 
66.1 


9,690 
9,820 
26.5 


As  the  reader  will  observe,  the  amomit  of  control  of  various  sorts  is 
given  in  the  above  table  with  reference  to  areas  in  square  inches  upon  the 
map  as  published.  It  is  given  in  these  terms  in  order  to  eliminate  from  it 
the  question  of  scale. 

No  statistics  of  horizontal  control  are  given  for  the  areas  surveyed  in 
Wisconsin,  Illinois,  and  Kansas,  because  most  of  it  is  furnished  by  the 
surveys  of  the  General  Land  Office,  and  therefore  the  presentation  would 
be  but  a  partial  one. 

There  are  two  general  methods  for  location  of  stations  and  of  minor 
points  for  the  coiTection  of  the  sketch,  the  one  by  angular  measurements 
(triangulation),  the  other  by  measurement  of  directions  and  distances, 
or  what  is  known  popularly  as  the  traverse  or  meander  method.  In  ordinary 
practice,  work  may  be  done  by  either  of  these  two  methods,  or  they  may 
be  used  in  conjunction.  The  former  of  the  two  methods  may  be  carried  on 
with  the  plane  table,  various  forms  of  the  theodolite,  with  a  compass,  or, 
indeed,  with  an  angle-reading  instrument.  The  latter  method  may  be  car- 
ried on  with  the  same  instruments,  supplemented  by  various  forms  of  odom- 
eters, chain,  steel  tape,  stadia,  etc.,  for  the  measurement  of  distance.  The 
first  method,  whenever  it  can  be  used  economically,  is  the  most  accurate, 


METHODS  OF  CONTEOL.  13 

and  is,  as  a  rule,  the  most  rapid,  and  the  locations  are  likely  to  be  of  the 
greater    service    and  distributed   most    uniformly.     It  can  be  used  eco- 
nomically where    the    country  presents    more  or   less  relief,  and  where 
points  for  location,  either  natural  or  artificial,  exist  in  sufficient  numbers 
and    are  well  distributed.     These   conditions  are   satisfied  almost  every- 
where in  the  western  mountain  regions,  where  mountain  peaks,   summits 
of  hills,  plateau    points,  buttes,  etc.,    furnish  an    abundance  of    natural 
points   for  stations   and   locations.     It   can   be   used,    to   a   considerable 
extent,  though  not  with  the  same  ease  or  economy  in  the  Appalachian 
mountains;  but  in  this  region  it  is  necessary  to  supplement  it  extensively 
by  traverse  lines,  especially  in  tracing  the  courses  of  streams  in  the  valleys. 
It  can  be  used,  too,  in  the  hill  country  of  New  England,  where  objects  of 
culture,  such  as  churches,  houses,  etc.,  furnish  an  abundance  of  signals.    On 
the  other  hand,  throughout  the  whole  extent  of  the  Atlantic  and  the  Gulf 
plains,  where  the  country  is  level  or  nearly  so,  and  is  covered  with  forests, 
the  tra,verse  method  of  surveying  must  be  resorted  to.     This  is  a  country 
devoid  of  sharp  natural  objects,  a  country  in  which  extended  views  can  not 
be  obtained.     The  only  economical  way,  therefore,  of  proceeding,  is,  start- 
ing from  some  point  located  by  the  triaugulation,  to  carry  a  line  of  stations, 
connected  together  by  distance  and  direction  measurements,  until  the  line 
checks  upon  a  second  triangulation  point.     For  many  reasons,  this  method 
of  obtaining  locations  is  inferior  to  the  former.     It  is  inferior  not  only  in 
accuracy,  but  in  the  facilities  which,  as  carried  out,  it  affords  for  sketching 
the  country,  and  it  should  be  so  regarded,  and  should  be  adopted  only  when 
it  becomes  necessary,  or  when  the  former  method  can  not  be  appHed  eco- 
nomically.    For  convenience,  traverse  lines  are  generally  run  along  the 
roads  or  trails,  and  thus  the  best  points  for  commanding  views  of  the  country 
are  avoided  rather  than  sought.     Being  practically  confined  to  the  roads, 
there  is  danger  that  the  topographer  neglects,  in  a  greater  or  less  measure, 
the  areas  lying  between  them.     On  account  of  the  errors  incident  to  run- 
ning a  traverse  it  is  necessary  that,  in  this  class  of  work,  frequent  locations 
be  made  by  triangulation  for  checking  and  thereby  eliminating  its  errors. 
The  locations  dealt  with  in  the  above  table  fall  into  one  or  the  other 
of  these  two  classes.     Locations  by  triangulation  are  of  much  greater  value 


14  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

than  those  by  traverse.  As  a  rule,  they  are  selected  points  chosen  because 
each  controls  positions  in  a  certain  area.  On  the  other  hand,  traverse  loca- 
tions are  not,  as  a  rule,  chosen  for  then  control  value,  but  only  for  inter^dsi- 
bility  on  roads.  Furthermore,  the  great  majority  of  traverse  stations  are  of 
no  service  whatever  beyond  carrying  the  line  forward,  so  that  in  estimating 
the  total  amount  of  control  in  a  certain  area  where  the  control  is  made  up 
in  whole  or  in  part  of  traverse  lines,  less  weight  should  be  given  to  them 
than  to  locations  by  triangulation. 

The  third  element  of  accuracy,  the  distribution  of  locations,  is  a  point 
concerning  which  it  is  equally  difficult  to  speak  definitely.  Other  things 
being  equal,  the  disti'ibution  should  be  uniform  over  the  area,  but  it  will 
necessarily  vary  with  the  character  of  the  surface.  The  accompanying 
diagram  shows  the  amount  and  distribution  of  control  in  a  typical  piece  of 
work.  In  general,  in  the  mountain  regions,  locations  by  angular  measure- 
ments are  frequent  and  accompany  the  ranges  or  ridges,  and  such  locations 
are  few  in  number  in  the  valleys,  being  supplemented  there  by  traverses. 

The  fourth  of  the  elements  of  the  correctness  of  the  map  depends  upon 
the  artistic  sense  of  the  topographer,  upon  his  ability  to  see  things  in  then- 
proper  relation,  and  his  facility  in  transferring  his  impressions  to  paper. 
This  is  by  far  the  most  important  and  the  most  difficult  to  meet. 

The  education  of  the  topographer,  therefore,  consists  of  two  parts — the 
mathematical  and  the  artistic.  The  first  may  be  acquired  largely  from 
books,  and  this  book  knowledge  must  be  supplemented  by  practice  in  the 
field.  The  second,  if  not  inherited,  can  be  acquired  only  by  long  experi- 
ence in  the  field,  and  by  many  can  be  acquired  only  imperfectly.  In  fact, 
the  sketching  makes  the  map,  and,  therefore,  the  sketching  upon  the  Oeo- 
logical  Survey  is  executed  by  the  best  topographer  in  the  party,  usually  its 
chief,  whenever  it  is  practicable  to  do  so. 


BUCKHANNON,  W.  VA,,  SHEET. 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH   XXII.     PL.  II. 


8 

,^>^      (^-         '^°^' 

y-  *  '^/r-^ 

s 

s 

PI 

Diagram.  sliCf\Fm^  diartxitTitLon  o£  control  work 

Statute  Miles 


Main  and.  Secotndanrr  stalioxis. 
inters  ecti-ons  firom.  stalians  . 


Intersec'tiums  £:onL  ■trar-erse . 
Traversed  Roads  or  Trails 


'^' 


CHAPTER     II. 

CLASSIFICATION  OF  WORK. 

The  Avork  involved  in  making  a  map  usually  comprises  several  opera- 
tions, which  may  in  practice  be  more  or  less  distinct  from  one  another. 
They  are  enumerated  as  follows: 

First— The  location  of  the  map  upon  the  earth's  surface,  by  means  of 
astronomic  observations. 

Second. — The  horizontal  location  of  points. 

This  is  usually  of  thi-ee  grades  of  accuracy,  primary  triangulation,  or 
primary  traverse,  in  cases  where  triangulation  is  not  feasible;  secondary 
triangulation  for  the  location  of  numerous  points  within  the  primary  triangu- 
lation; and  ordinary  traverse,  for  the  location  of  details. 

XJiircl— The  measurement  of  heights,  which  usually  accompanies  the 
horizontal  location,  and  which  may,  similariy,  be  divided  into  three  classes, 
in  accordance  with  the  degree  of  accuracy. 

Fourth. — The  sketching  of  the  map. 

Nearly  all  of  the  geometric  work  of  the  Survey,  the  work  of  location, 
is  executed  by  five  instruments. 

Theodolites,  of  a  powerful  and  compact  form,  used  in  the  primary 

control. 

Plane  tables,  with  telescopic  alidades  of  the  best  type,  used  for  second- 
ary triangulation  and  height  measurements. 

Plane  tables,  of  crade,  simple  form,  with  ruler  ahdades,  used  for 
ti-aversing  and  minor  triangulation. 

Odometers,  for  measuring  distance. 

Aneroids,  for  the  measurement  of  details  of  heights. 


1(^  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

With  these  instruments  nine-tenths  of  the  work  is  done,  and  these 
instruments  will  be  described  in  their  proper  places  with  such  fullness  of 
detail  as  seems  necessar)^ 

Other  instruments,  such  as  transits,  surveyors'  theodolities,  compasses, 
wye  levels,  hand  and  Abney  levels,  telemeters,  chains,  tapes,  and  mercurial 
barometers,  are  occasionally  used.  Most  of  these  instruments,  which  are 
commonly  figured  and  described  in  all  works  on  survejang,  are  assumed  to 
be  well  known  to  the  readers  of  this  manual  and  will  therefore  receive  no 
special  attention. 

ASTRONOMIC  DETERMINATIONS  OF  POSITION. 

The  object  of  astronomic  determinations  of  position  is  to  locate  the 
map  upon  the  earth's  surface.  They  are  made  also  for  the  purpose  of 
checking  and  correcting  positions  determined  by  primary  triangulation  and 
primary  traverse. 

With  regard  to  the  checking  of  the  primary  triangulation  by  astronomic 
determinations,  it  should  be  understood  that  in  the  case  of  a  single  determi- 
nation -the  work  by  triangulation  is  far  more  accurate  than  by  astronomic 
determinations,  even  when  made  iinder  the  best  of  circumstances.  It  is, 
therefore,  desirable  to  introduce  checks  of  this  kind  upon  primary  triangu- 
lation only  when  the  latter  has  been  carried  for  a  long  distance,  200  or  300 
miles,  for  instance,  in  the  course  of  which  it  may  have  accumulated  errors 
greater  than  those  incident  to  astronomic  work. 

The  case  is  different  with  primary  traverse.  The  great  number  of 
courses  required  in  this  work  affords  an  opportunity  for  the  accumulation 
of  error  much  greater  than  is  the  case  with  triangulation,  and  consequently 
it  is  desirable  to  introduce  more  frequent  checks  in  this  work.  It  may  be 
said  that,  in  general,  such  work  should  be  checked  at  every  100  miles. 

As  was  suggested  above,  the  best  astronomic  determinations  are  none 
too  good  for  the  control  of  maps.  Indeed,  certain  errors  hicident  to  this 
work,  some  of  which  as  yet  can  not  be  corrected,  may  be  of  magnitude 
sufficient  to  show  upon  the  scale  of  the  map.  It  is  necessary,  therefore,  in 
these  determinations  to  use  the  best  instruments    and  the  most  refined 


ASTRONOMICAL  DETEEMESTATIONS.  17 

methods  known  to  modern  science,  in  order  to  reduce  all  avoidable  errors 
to  a  minimum. 

Whatever  determinations  have  been  made  by  the  U.  S.  Coast  and 
Geodetic  Survey,  the  United  States  Lake  Survey,  or  the  Mississippi  River 
Commission,  whether  by  astronomical  work  or  by  triangulation,  these  posi- 
tions may  be  utilized  for  the  above  purposes. 

DEFmiTIONS. 

Sidereal  time  is  the  time  indicated  by  the  stars,  a  sidereal  day  beinp^ 
the  time  which  elapses  between  two  passages  of  the  vernal  equinox  across 
the  meridian.  Solar  or  apparent  time  is  the  time  measured  by  the  sun's 
apparent  movement  or  the  revolution  of  the  earth  with  reference  to  the  sun, 
and  since  the  earth  revolves  at  a  differing  rate  in  different  portions  of  its 
orbit,  the  solar  days  are  not  of  equal  length.  A  mean  day  is  the  average 
solar  day;  mean  time  differs  from  solar  time  by  an  amount  which  varies 
with  the  time  of  year,  and  which,  under  the  name  of  "  equation  of  time,"  is 
given  in  the  Nautical  Almanac.  Mean  time  differs  from  sidereal  time  by 
about  a  day  in  the  com'se  of  a  year,  or  about  four  minutes  in  each  day; 
the  mean  day  being  longer  than  the  sidereal  day.  To  convert  a  given  date 
of  mean  time  into  sidereal  time  it  is  necessary  to  obtain,  from  the  Nautical 
Almanac,  the  sidereal  time  at  noon  immediately  preceding  the  date  in  ques- 
tion. Then  the  interval  after  noon,  expressed  in  mean  time,  is  converted 
into  sidereal  time  by  table  xxxii  in  this  volume,  and  the  result  added  to  the 
sidereal  time  of  mean  noon.  Local  time,  whether  sidereal,  solar,  or  mean, 
is  the  time  of  the  locality  as  distinguished  from  the  time  of  any  other 
locality.  It  must  be  distinguished  from  railroad  time,  which  is  the  local 
time  only  of  certain  meridians. 

The  right  ascension  of  the  sun  or  a  star  is  the  sidereal  time  which  has 
elapsed  between  the  passage  of  the  vernal  equinox  and  the  star  across  the 
meridian.     It  is  commonly  expressed  in  hours,  minutes,  and  seconds. 

Declination  is  the  angular  distance  of  a  heavenly  body  north  or  south 
of  the  equator.     It  is  plus  when  north  and  minus  when  south  of  the  equator. 

The  zenith  distance  of  a  heavenly  body  equals  its  declination,  minus 
the  latitude  of  the  place  of  observation. 

Latitude  is  determined  by  what  is   known  as  Talcott's  method,  by 

MOKf  XXII 2 


18 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


measuring  the  differences  of  zenith  distance  at   cuhnination   of  two  stars 
which  cuhninate  on  opposite  sides  of  the  zenith. 

Longitude  is  determined  by  telegraphic  comparison  of  local  time  at 
two  stations,  the  longitude  of  one  of  which  is  known.  This  involves  the 
determination  of  the  errors  of  the  clocks  or  chronometers  used,  which  is 
done  by  observation  of  transits  of  stars  across  the  meridians  of  the  places  of 
observation. 

ASTRONOMICAL  TRANSIT  AND  ZENITH  TELESCOPE. 

A  single  instrument  is  used  for  the  determination  both  of  latitude  and 

time.  This  is  a  combination  of 
the  transit  aiid  zenith  telescope. 
The  instruments  in  use  upon  the 
Geological  Survey  were  made  by 
Saegniuller  and  embody  the  latest 
improvements  in  these  combined 
instruments.  One  of  them  is  fig- 
ured herewith.  The  circular  base 
rests  upon  three  leveling  screws. 
Upon  this  circular  base  the  whole 
instrument  can  be  made  to  re- 
volve when  using  it  as  a  zenith 
telescope  A  circle  is  graduated 
around  the  base,  having  a  microm- 
eter screw  for  slow  motion,  for 
making  settings  and  adjusting  the 
instrument  in  azimuth.  The  frame 
of  the  instrument  is  cast  in  one 
piece,  and  the  standards  are  hol- 
low in  order  to  reduce  the  weight 
of  the  upper  part  of  the  instrument. 
The  telescope  has  a  focal  distance 
of  27  inches  and  a  clear  aperture 

Fig.  1.— Astronomical  transit  and  zenith  telescope.  pf     2.5     inchcS.         ItS      magnifying 

power  with  diagonal  eyepiece  is  74  diameters.     The  length  of  the  axis  of 


ASTEONOMICAL  DETEEMINATIONS.  19 

the  telescope  is  16  inches.  For  use  as  a  zenith  telescope,  the  telescope  is 
equipped  with  a  vertical  circle  reading  by  vernier  to  20  seconds,  attached 
to  which  is  a  delicate  level.  In  the  focus  of  the  object-glass  there  is,  besides 
the  ordinary  reticule  for  use  in  transit  work,  a  movable  thread,  which  is 
moved  by  means  of  a  micrometer  screw,  by  which  measurements  of  differ- 
ences of  zenith  distances  are  made.  It  is  furnished  with  direct  and  diagonal 
eyepieces,  the  latter  of  which  is  commonly  used  in  astronomical  work. 

For  use  as  a  transit  instrument,  the  telescope  is  equipped  with  a  deli- 
cate striding  level  for  measuring  the  inclination  of  the  pivots,  and  a  revers- 
ing apparatus  for  turning  the  telescope  in  the  wyes.  The  reticule,  as  the 
stationary  threads  in- the  focus  of  the  instrument  are  called,  consists  of  five 
threads  for  observing  the  transits  of  stars.  The  reticule  is  illuminated  by 
means  of  bull's-eye  lamps,  the  light  from  which  comes  through  the  hollow 
axis  of  the  telescope  and  is  reflected  by  a  mirror  placed  at  the  intersection 
of  the  telescope  with  its  axis. 

CHRONOGRAPH. 

The  chronograph  is  used  for  the  purpose  of  recording  the  time  of 
transits  of  stars  as  observed  with  the  transit  instrument.  It  may  be  popu- 
larly characterized  as  an  instrument  for  measuring  time  by  the  yard.  It 
consists  essentially  of  a  drum,  upon  which  is  wound  a  strip  of  paper,  and 
which  is  kept  in  revolution  by  .a  train  of  clockwork  controlled  by  an  escape- 
ment. A  pen  carried  upon  a  small  car,  which  is  moved  very  slowly  in  a 
direction  parallel  to  the  axis  of  the  cylinder,  traces  a  spiral  line  upon  the 
paper  on  the  drum.  This  pen  is  held  in  place  by  a  magnet,  which  is  carried 
upon  the  car,  and  as  long  as  the  current  from  the  battery  passes  through 
the  coil  and  thiis  holds  the  armature  the  pen  traces  an  unbroken  spiral  line. 
If  the  current  is  suddenly  broken  and  restored,  the  armature  is  set  fi-ee  for 
an  instant  and  a  jog  is  made  in  the  line  traced.  The  battery  commonly 
used  in  connection  with  this  outfit  is  the  ordinary  zinc,  copper,  and  sulphate 
of  copper  battery,  of  which  four  cells  are  usually  required.  The  ordi- 
nary dry  battery  can  also  be  used  and  is  much  more  convenient.  With  this 
apparatus  break-circuit  chronometers  are  used.  These  difi^'er  from  ordinary 
chronometers  in  the  fact  that  they  are  arranged  to  break  an  electric  circuit 


20 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


automatically  at  regular  intervals.  Those  in  use  upon  the  Geological  Sur- 
vey break  the  circuit  every  two  seconds,  and  the  end  of  the  minute  is  indi- 
cated by  breaking  at  the  fifty-ninth  as  well  as  the  fifty-eighth  and  sixtieth 
seconds.  When  one  of  these  chronometers  is  coimected  with  a  battery  and 
a  clu'onograph  is  put  in  the  same  circuit,  the  beginning  of  every  even  sec- 
ond is  recorded  upon  the  chronograph  by  a  jog  on  the  paper,  and  the  dis- 
tance between  the  jogs  in  each  case  represents,  therefore,  two  seconds.  The 
observer  at  the  instrument  is  provided  with  a  telegraph  key,  which  may  also 
be  put  in  the  circuit  with  the  clu'onometer  and  chronograph,  and  as  a  star 


Fig.  2. — Chronograph. 

near  the  meridian  crosses  a  thread  in  the  telescope  he  records  that  fact  by 
pressing  on  the  key,  which  makes  a  record  upon  the  chronograph  along 
with  the  record  of  the  chronometer. 

An  illustration  of  the  form  of  clu-onograph  in  use  upon  the  Geological 
Survey  is  shown  upon  Fig.  2. 

FIELD    WORK. 

Since  the  observations  for  latitude  and  longitude,  though  different, 
are  made  with  the  same  instrument,  at  the  same  time,  and  by  the  same 
party,  certain  parts  of  the  work  apply  equally  to  both  determinations  and 
may  be  described  once  for  all. 


ASTEONOMICAL  DETEEMINATIONS.  21 

lu  the  selection  of  a  station,  care  must  be  taken  to  avoid  a  locality 
where,  for  any  cause,  the  ground  is  liable  to  be  seriously  jarred,  as,  for  in- 
stance, proximity  to  a  railroad  track  or  to  a  street  over  which  heavy 
wagons  pass.  It  should  have  a  clear  view  from  the  southern  horizon  through 
the  zenith  to  the  northern  horizon.  It  is  desirable  to  locate  at  a  convenient 
distance  from  a  telegraph  station,  as  it  is  necessary  to  bring  a  wire  in  from 
siTch  station  for  the  purjDOse  of  comparing  chronometers.  If  possible,  the 
station  should  be  selected  upon  a  public  reservation,  in  order  that  the  per- 
manence of  the  monument  marking  the  spot,  which  is  to  be  erected,  may 
be  assured.  But,  in  any  event,  one  should  avoid  a  locality  in  which  such  a 
monument  is  likely  to  be  disturbed. 

The  support  of  the  instrument  should  consist  of  a  brick  pier  sunken 
fully  three  feet  in  the  ground  and  rising  above  it  to  the  requisite  height. 
Upon  this  should  be  placed  for  the  immediate  support  of  the  instrument,  a 
block  of  stone  well  set  in  mortar.  The  cln-onograph  may  be  set  up  on  an 
ordinary  table.  Over  all  should  be  erected  a  wall  tent  with  a  slit  closed 
by  flaps  in  the  roof,  which  can  be  opened  when  observing.  The  instrument 
is  set  up  upon  the  pier,  collimated,  leveled,  and  the  verticality  of  the 
threads  tested  as  accurately  as  possible,  and  is  then  pointed  upon  the  pole 
star.  This  places  it  somoAvhere  near  the  meridian.  Then  taking  the  time 
of  transit  of  a  star  which  culminates  close  to  the  zenith,  and  comparing  this 
time  with  the  right  ascension  of  the  star,  a  sufficiently  close  approximation 
to  the  clock  error  is  obtained  for  use  in  placing  the  instrument  in  the  meri- 
dian. The  instrument  is  then  turned  in  azimuth  to  point  upon  some  close 
circum-polar  star  approaching  upper  or  lower  culmination,  mo\ang  the  in- 
strument in  azimuth  with  the  tangent  screw  so  as  to  keep  the  star  under 
the  middle  wire  up  to  the  instant  of  culmination.  If  this  is  done  accurately 
at  the  first  attempt,  the  instrument  is  placed  nearly  in  the  meridian  and  is 
ready  for  work,  but  it  commonly  happens  that  more  than  one  trial  is 
required  before  the  meridian  is  reached.  In  any  case,  the  result  should  be 
verified  by  a  second  star,  before  proceeding  with  the  observations. 

OBSERVATIONS    FOR    LATITUDE. 

As  preliminary  to  this  work  it  is  necessary  to  prepare  a  list  of  pairs  of 
stars,  the  two  stars  of  each  pair  liaving  such  zenith  distances  that  they  will 


22  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

culminate  at  nearly  equal  distances  from  the  zenith,  one  to  the  north  and 
the  other  to  the  soiith  of  it.  Such  a  list  can  be  prepared  from  the  Saffbrd 
•Catalogue  of  the  Wheeler  Survey.  For  this  it  is  necessary  to  know  the 
approximate  latitude  of  the  station,  the  right  ascensions  and  the  declina- 
tions of  the  stars.  The  zenith  distance  of  a  star  is  equal  to  its  declination, 
minus  -the  latitude  of  the  place.  The  stars  of  each  pair  should  culminate 
within  a  few  minutes  of  one  another.  They  must  be  observed  consecu- 
tively, and,  therefore,  those  stars  should  be  selected  which  culminate  as 
nearly  as  possible  together,  leaving  only  a  sufficient  interval  of  time 
between  them  for  setting  the  instrument. 

Before  beginning  to  observe,  the  instrument  should  be  closely  coUimated 
and  di'awn  into  the  meridian. 

Upon  the  approach  of  the  first  star  of  the  pair  to  the  meridian,  the 
instrument  shoidd  be  set  for  it,  using  the  vertical  circle  for  that  purpose, 
and  setting  the  spirit  level  upon  the  vertical  circle  as  nearly  level  as  possi- 
ble. Then,  as  the  star  traverses  the  field  of  the  telescope,  keep  the  movable 
thread  in  the  reticule  upon  it  by  means  of  the  micrometer  screw  until  it 
crosses  the  middle  vertical  thread.  Then  read  and  record  the  micrometer 
and  the  two  ends  of  the  level  bubble.  Without  disturbing  in  the  slightest 
degree  the  setting  of  the  telescope,  turn  the  entire  instrument  180°  upon  its 
bed  plate,  when  it  will  point  north  of  the  zenith,  at  the  same  angle  that  it 
formerly  pointed  south,  or  vice  versa,  as  the  case  ma,y  be,  and  will  be  set 
for  the  other  star  upon  the  opposite  side  of  tl'e  zenith.  As  this  approaches 
culmination,  follow  it  with  the  micrometer  as  before,  until  it  reaches  the 
middle  thread;  then  record  as  before  the  readings  of  the  micrometer  and  of 
the  level,  whether  it  has  changed  or  not. 

This  constitutes  the  observations  upon  a  single  pair  of  stars.  For  the 
determination  of  latitude  twenty  such  pairs  of  stars  should  be  observed 
each  evening,  if  possible,  and  the  same  pairs  of  stars  should,  also  assuming 
it  to  be  possible,  be  observed  upon  other  evenings.  The  following  exam- 
ple, taken  from  observations  at  Rapid,  South  Dakota,  shows  a  portion  of 
the  star  list  and  the  form  of  record: 


ASTRONOMICAL  DETEEMINATIONS. 


23 


LATITUDE    DETERMINATION. 

List  of  Stars,  for  Observation  icith  Zenith  Telescope. 
[Station:    Eapid,  Sowth  Dakota.    Approximate  Latitude:  44°  05'.] 


Name  or 
number. 
Saftbrd's  Cat- 
alogue. 

Mag. 

Class. 

E.  A. 

Dec. 

Zen 

dist. 

Setting. 

7  LacertcE 

10  Lacertje  . . . 

4.0 
5.0 

6.5 
6.5 

6.5 
5.0 

6.0 

0.7 

5.6 
6.5 

4.5 
6.5 

A  A 

A  A 

B 

A 

C 
A 

A 
B 

A 
A 

A 
B 

h. 

22 
22 

22 

22 
22 

23 
23 

23 
23 

23 
24 

27 
34 

41 
47 

59 

08 
18 

42 
47 

52 
00 

49°  43' 
38    29 

45    37 
42    42 

38    42 
49    26 

56    34 
31     56 

67     12 
21     03 

24     32 
03     35 

5 

1 
1 

12 
12 

23 
23 

19 
19 

38'  is". 

36    S. 

32  N. 
23    S. 

23    S. 

21   N. 

29  If. 
09    S. 

07  N. 
02     S 

33  S. 

30  N. 

1    5     37  N. 
^    1     27  N. 
J    5    22   S. 
1 12    19  ST. 
I  23    05  N". 
|l9    31   S. 

1676      

1686 

1722     

Example  of  Record. 

[Station:  Rapid,  South  Dakota.    Date;   November  9,  1890.    Instmment:   Fautli  combined  transit  and  zenith  teleacop 
No.  534.    Obseiver:  S.  S.G.    Eecorder;  A.F.D.] 


Star  name  or 
number. 

N.or 
S. 

— « 

Microm- 
eter 
reading. 

Diff. 

Level. 

(N+S) 
-(N'+S') 

Remarks. 

N. 

S. 

7  Lacertas 

lOLaoerta) 

N. 
S. 

N. 
S. 

S. 

N. 

N. 
S. 

N. 
S. 

S. 

N. 

Eev. 
26. 256 
24.  052 

30. 432 
20.  095 

25.164 
26.  703 

32.  214 
16. 033 

26.  656 
17.684 

25. 345 
23. 722 

Sev. 
—2.204 

-10.337 
+1.  539 

-16. 181 
-8.  972 
+1.  623 

Div. 
39.9 
26.5 

42.0 
21.9 

14.1 
38.1 

37.5 
19.9 

51.0 
17.0 

18.0 
36.0 

Div. 
16.7 
49.7 

18.7 
45.0 

37.6 
15.0 

14.1 
43.1 

28.0 
39.6 

40.9 
13.2 

Biv. 
+56.  6 
—76.2 
-19.6 

+60.7 
—66.9 
-  6.2 

-51.7 
+53.1 
+  1.4 

+51.6 
-63.0 
-11.4 

+79.0 
—56.6 
-22.4 

-58.9 
+49.2 
-  9.7 

Faint. 
Distinct. 

Faint 

1686         

1722 

REDUCTION    OF    LATITUDE    OBSERVATIONS. 

Before  proceeding  with  the  reduction  of  latitude  observations,  it  is  nec- 
essary to  investigate  tlie  constants  of  the  instrument,  to  ascertain  tlie  value 
of  a  division  of  the  latitude  level,  and  of  a  division  of  the  head  of  the 
micrometer  screw. 

The  value  of  a  division  of  the  head  of  the  micrometer  screw  is  measured 


24  A  MANUAL  OP  TOPOGEAPHIO  METHODS. 

by  observing-  the  transits  of  some  close  circumpolar  star,  when  near  elong-a- 
tion,  across  the  movable  tlu-ead,  setting  the  thread  re^oeatedly  at  regular 
intervals  in  advance  of  the  star,  and  taking  the  time  of  its  passage,  with  the 
reading  of  the  micrometer.  The  precaution  should  be  taken  to  read  the 
latitude  level  occasionally  and  correct  for  it  if  necessary.  This  correction, 
which  is  to  be  applied  to  the  observed  time,  is  equal  to  one  division  of  the 
level,  in  seconds  of  time,  divided  by  the  cosine  of  the  declination  of  the 
star  and  multiplied  by  the  level  error,  the  average  level  reading  being 
taken  as  the  standard. 

The  time  from  elongation  of  the  star  requires  a  correction  in  order  to 
reduce  the  curve  in  which  tlie  star  apparently  travels  to  a  vertical  line. 
The  hour  angle  of  the  star  is  first  obtained  from  the  equation, 

cos  t^  zz  cot  d  tan  q), 
S  being  the  star's  declination  and  q>  the  latitude. 

.  The  clu'onometer  time  of  elongation,  To  zz  a  —  t^  —  St,  a  being  the 
right  ascension  of  the  star  obtained  from  the  JsTautical  Almanac,  and  U  the 
error  of  the  chi'onometer. 

Ha^ang  thus  obtained  the  cln-onometric  time  of  elongation,  the  correc- 
tion in  question  is  obtained  from  the  observed  interval  of  time  of  each  ob- 
servation before  or  after  elongation,  from  tables  in  Appendix  No.  14,  U.  S. 
Coast  and  Greodetic  Survey  Report  for  1880,  pp.  58  and  59.  A  discussion 
of  tliis  subject  will  be  found  in  the  appendix  above  referred  to,  and  in 
Chauvenet's  Practical  Asti-onomy,  vol.  ii,  pp.  360  to  364. 

The  times  of  observation  thus  corrected  for  level,  and  distance  from 
elongation,  are  then  grouped  in  pairs,  selected  as  being  a  certain  number  of 
revolutions  of  the  micrometer  apart,  and  the  time  intervals  between  the 
members  of  each  pair  obtained.  The  mean  of  these,  divided  by  the  sum  of 
revolutions  which  separate  the  members  of  each  pair,  is  yet  to  be  corrected 
for  differential  refraction,  which  is  derived  from  the  following  equation: 

Ref.  —  bl"  .7  sin  B  sec^  Z. 
R  being  the  value  of  a  division  of  the  micrometer  and  Z  the  zenith  distance  of 
the  star.  Four-place  logarithms  are  sufficient  for  computing  this  correction, 
as  it  is  small.  Below  is  given  an  example  of  record  and  computation  of  the 
value  of  a  revolution  of  the  micrometer  of  combined  instrument  No.  534, 
one  of  the  two  in  possession  of  the  Greological  Survey. 


TABLE  OF  DETEEMINATION. 


25 


fei=;; 


oi 

'if 

m.       s. 

15    37.7 

36.6 

34.4 
39.2 
40.2 
36.9 
36.8 
37.5 
36.2 
29.6 
33.1 
36.5 
31.3 
36.2 
15   35.80— mean 

o 

"1  ,  ^ 

S  i 

O        CO 

II 

.    2! 

1 

? 

1i 

ill 

tbiifc 

i 

1 

ibifc 

=.     '^      fc.       : 

1        t^       K         ■ 

;     "    b     ■ 
g    3    3     ; 
i    \          • 

1 1     i 

ll 

s 

£  0.-I 

'■  t-^  d  d  d  rH 

■  OOOOClOiH 

rHD^ddj^dddt^g 

co-*in!DOc-oo«or:; 

:g3SSS3SSS5S 

!cgC0'*lOt0l>Q0C0C5Or-* 
|OOOOOOOOOrH.-( 

i1- 

■*CO 

dddd    ■ 

+-i-4-h  : 

•  W<MMt--(iH 

■  ddddd 

.-lOOOOOOOOOOOOOOO  i-Hy^  1-4  CQ  IM  CO 

ddddd dddd ddddd ddddd odd 
++     ■              +1                           1   1   1   1   !   1 

So 
§g" 

"7" 

j  ^  «  o"  CO  a5 

o6i-JMdcddd-*t--d 

COTj<ir5lOOrH(MCOr-!0 

1  + 

■  j^'*coirid'*'*i>odd-^ 
.in^cowi-H'HOLrs^'S'CQ 

S  g  i  s 

inco      ,-,ccinMocoinoor-4i-HOOCiC50sc:C!CiC30Jooooi>oo»Oin'*ii«MrHomcDoa 

.^^        rHOdoOOOOr^rHrHrHOOodoOOOOOOdoOOOOOOOOOO 

++    +++++I  1  1  1  M  .  1  1  1  M  1  1  1  1  1  1  1  1  M  1  1  1  :  I+++ 

J! 

+ 

o 
1 

■d 

:  1 

1   : 

° 

;° 

o 

+ 

•3     «^    ■ 
o      + 

Is 

si 

s 

:S 

s  • 

o 

s 

is 

i 

II 

5 

3J 

3 

- 

;t^ 

S  ^ 

o 

b 

s  I 

^ 

t-" 

3 

1  ^ 

3; 

i 

;b^ 

"'     : 

•enoiitix 

•OAOI 1918 

-raojotit 

om      omotooinomoiaomomooomoiooifloirioioomomomomo        | 
T-H  d      d  d  d  oc  cc  t>^i>  d  d  lOiri -^ -*  CO  co^  M  r-H  i-id  d  d  d  c6c«5  c^t- d  diriifi**-^  coco        • 

Time  of  observa- 
tion (recorded 
on  chronograph 
sheet) . 

•^8    |l            ~"                     •                   " 

26 


A  MANUAL  OF  TOPOGEAPHIG  METHODS. 


The  value  of  a  division  of  the  level  is  commonly  measured  with  a  level 
trier.  The  latitude  level  may,  however,  be  easily  measured  by  means  of 
the  micrometer,  the  value  of  a  revolution  of  that  being  obtained  by  the  fol- 
lowing method: 

Point  the  telescope  upon  some  well-defined  terrestrial  mai-k  and  set 
the  level  at  an  extreme  reading  near  one  end  of  the  tube.  Set  the  movable 
thread  upon  the  object  and  read  the  micrometer  and  the  level. 

Now  move  the  telescope  and  level,  until  the  bubble  is  near  the  other 
^nd  of  the  tube.  Again  set  the  movable  tlu-ead  upon  the  object  and  again 
read  both  micrometer  and  level.  It  is  evident  that  the  micrometer  and  the 
level  have  measured  the  same  angle,  and  that  the  ratio  between  these  read- 
ings equals  that  between  h  revolution  of  the  micrometer  and  a  level  division. 

An  e:5ample  illustrative  of  this  is  appended. 

Determination  of  I'alue  of  1  division  of  latitude  level  No.  534. 
[By  comparison  with  micrometer  screw  534  ] 


Microme- 
ter. 

Level. 

Diffei 

enCB. 

aa. 

ab. 

N. 

S. 

Microm. 

Level. 

r. 

8.025 
8.508 

d. 

47.3 
20.7 

d. 
29.2 
02.7 

b. 
d. 
48.3 

a. 
d. 
26.55 

704.9 

1283. 

8.509 
7.984 

18.9 
49.8 

01.0 
31.0 

52.5 

30.45 

927.  2 

1599. 

8.511 
8.045 

18.5 
47.2 

00.6 
29.1 

46.6 

28.60 

818.0 

1333. 

9.076 
8.604 

18.7 
46.0 

00.8 
28.0 

47.2 

27.25 

742.6 

1286. 

9.442 
9.  009 

23.7 

48.0 

06.0 
30.0 

43.3 

24.15 

583.2 

1046. 

10.  055 
9.574 

21.8 
48.0 

04.0 
30.1 

48.1 

26.15 

683.8 

1258. 

10.661 
10.  212 

24.0 
50.7 

06.1 
83.0 

44.9 

26.80 

718.2 

1203. 

11.771 
1].252 

18.3 
48.3 

00.7 
31.9 

51.9 

30.60 

936.4 

1588. 

12.  328 
11.872 

20.0 
46.1 

02.3 
28.5 

45.  C 

26.15 

683.8 

1192. 

12.  869 
12.  438 

22.2 
47.7 

04.6 
30.0 

43.1 

25.45 

647.7 

1097. 

13.  468 
13.080 

23.0 
44.5 

05.3 
26.9 

38.8 

2L55 

464.4 

836. 

14. 146 
13.702 

20.1 
45.4 

02.4 
27.8 

44.4 

25.35 

642.6 

1125. 

14.  758 
14.  282 

Sum. 

22.3 
48.6 

04.8 
31.0 

47.6 

26.25 

689.1 

1249. 

9241.9 

16095. 

log 16095.  =4.20669. 

A.  C.  log 9241.9  =  6.03424. 

log  1  Div.  Micrometer =9.87966. 

IDiv.  level =1".320  log.  =0.12059. 


LATITUDE  DETERMINATION.  27 

Following  the  determination  of  the  constants  of  the  instrument  used, 
the  next  step  is  to  obtain  the  apparent  declinations  of  the  stars  used.  When- 
ever possible,  these  should  be  taken  from  the  Nautical  Almanac  or  the 
Berliner  Jahrbuch.  In  other  cases  they  must  be  computed.  The  positions 
of  stars  are  given  in  Safford's  Catalogue,  for  the  epoch  1875.0,  together  with 
the  annual  precession  and  proper  motion.  The  declinations  there  given 
should  be  revised  by  the  aid  of  more  recent  catalogues,  particularly  with 
reference  to  stars  of  class  C.  The  annual  precession  and  proper  motion 
multiplied  by  the  number  of  years  which  have  elapsed  and  applied,  together 
with  the  effect  of  secular  variation  in  precession,  give  the  declination  at  the 
beginning  of  the'year.  Further  corrections  to  bring  the  positions  down  to 
the  date  of  observation  are  expressed  by  the  symbols  Aa',  Bh',  Cc',  Dd'. 
Logarithms  of  a',  b',  c',  d'  are  given  in  Safford's  Catalogue,  and  A,  B,  C,  and 
D  are  given  in  the  Nautical  Almanac.  A  slight  additional  correction,  also, 
is  to  be  made  for  proper  motion,  for  the  elapsed  portion  of  the  year.  This 
reduction  is  illustrated  below. 

LATITUDE   DETERMINATION. 

Example  of  reduction.    Computation  of  apparent  declination  of  star  1539. 

[From  Safford's  Catalogue,  p.  40.] 

Star   No.  1539 

Yr. 

(1890-1875)    X  18. 87=  +4    43. 05  =  Precession  for  15  years. 

15X— .03=  —0    00. 45  =  Proper  motion  for  15  years. 

+  0    00. 07  =  Seciilar  variation  in  precession. 


Declination, 1875.0     p^eees"sl. 
45    33    29.20             ,   ^'J  g, 

Propter 
tnotioc. 

—.03 

45    38     11. 87  =  Declination  1890. 
+  9.38=  Aa' 

—  0.  78  =  B  b' 
+  6.88=C<-' 
+  10. 36=D(1' 

—  0.03  =  Proper  motion,  Jan.  1— Nov.  9,  1890. 


45    38    37. 48  =  Declination  Nov.  9,  1890. 


0. 9723 
a'  =  +  9. 38 


With  all  this  preliminary  work  done,  the  reduction  proper  of  latitude 
observations  is  comparatively  a  simple  matter.  Grrouping  the  observations 
by  pairs,  the  mean  declination  of  each  pair  is  obtained,  the  corrections  for 


28 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


difference  of  niicrometei'  readings  and  levels  are  applied,  witli  a  small  cor- 
rection for  differential  refraction,  and  the  result  is  the  desired  latitude. 

Following  is  an  example  of  the  reduction  of  six  pairs  of  stars  observed 
for  latitude  at  Rapid,  South  Dakota: 

LATITUDE    DETERMINATION. 

Example  of  Eediiction. 
[Station:  Eapid,  Soutli  Dakota.    November  9,  1890.    Half  Eev.  Micrometer=37.900.    One  Div.  Level-.:1.33.] 


Date. 

Star  num- 
bers. 

i. 

i2 

Hh  +  ii) 

Corrections. 

Latitude 
n. 

Weight 
P- 

p.  n. 

Microm. 

Level. 

Eefr. 

Not.  9. 

JTLacertandJ 
\  lOLacert.  S 

« 

42 

87.33 

38 

29 

04.60 

44  06  15.97 

—  1 

23.53 

-6.51 

—.03 

44   04   45.90 

.98 

5.78 

1539    1551 

45 

38 

37.  4S 

42 

44 

04.63 

11  21.06 

-  6 

31.77 

—2.06 

—.11 

47.12 

.90 

6.41 

1565    1579 

38 

43 

39.78 

49 

27 

41.04 

05  40.41 

-  0 

58.33 

+0.46 

—.03 

42.51 

.79 

1.98 

1600    1633 

56 

34 

06.66 

31 

55 

56.91 

15  01.78 

-10 

13.25 

-3.78 

-.19 

44.56 

.90 

4.10 

1676    16S6 

67 

12 

10.93 

21 

03 

54.02 

08  02.48 

-  3 

08.43 

-7.44 

-.07 

46.54 

.93 

6.08 

1702    1722 

21 

32 

09.04 

63 

35 

27.34 

03  48.19 

+  1 

01.51 

-3.23 

+.02 

46.  5U 

.90 

5.85 

5.40 

30.20 

November  9.    "Weighted  mean =44°  04'  45.59". 


OBSERVATIONS   FOR   TIME. 

With  the  transit  mounted,  leveled,  and  adjusted  in  the  meridian,  the 
chronograph  set  up  and  rumaing  and  connected  in  a  circuit  with  the  battery, 
and  the  chronometer  and  observing  key  connected  in  the  same  circuit  the 
observer  is  prepared  to  begin  time  observations. 

The  list  of  stars  which  should  be  used  is  that  given  in  the  Berliner 
Jahrbuch  as  the  list  is  fuller  and  more  accurate  than  that  in  any  other  cat- 
alogue which  gives  day  places.  Stars  should  be  so  selected  north  and  south 
of  the  zenith  tliat  the  azimuth  errors  will  balance  one  another  as  nearly  as 
possible,  as  is  explained  hereafter.  On  the  approach  of  the  selected  star  to 
the  meridian,  the  telescope  is  set  by  means  of  the  vertical  circle  upon  the 
altitude  of  the  star  above  the  horizon,  deduced  from  the  declination  and  the 
latitude.  As  the  star  crosses  each  tln'ead  in  the  reticule,  the  fact  is  recorded 
by  pressing  the  observing  key,  which  produces,  as  described  above,  a  record 
upon  the  chronograph  sheet.  In  this  way  four  time  stars,  as  stars  between 
the  equator  and  zenith  are  designated,  and  one  circumpolar  star,  or  a  star  so 
near  the  pole  that  it  is  constantly  in  sight,  should  be  observed.  Then  the 
telescope  should  be  reversed  in  the  wyes  and  a  similar  set  of  stars  observed. 


OBSEEVATIONS  FOE  TIME.  29 

« 

Between  observations  upon  any  two  stars  the  striding  level  sliotild    be 

placed  upon  the  pivots  of  the  instrument  and  readings  taken  to  ascertain 

the  departure  of  the  axis  from  a  horizontal  position. 

In  order  to  avoid  unequal  expansion  of  the  pivots  from  unequal  heat- 
ing, both  bull's-eye  lamps  must  be  lighted  and  placed  in  their  stands,  in 
order  that  both  pivots  may  be  equally  heated. 

After  the  comparison  of  chronometers  at  the  two  stations,  to  be  here- 
after described,  a  similar  set  of  stars  should  be  observed,  if  possible. 

EEDUCTION    OP   TIME    OBSERVATIONS. 

Certain  constants  of  the  transits  should  be  measured  before  proceeding 
with  the  reduction  of  time  observations.  The  value  of  a  division  of  the 
striding  level  should  be  measured  by  means  of  a  level  trier.  The  equatorial 
interval  of  time  between  each  of  the  threads  and  the  mean  of  all  the  threads 
should  be  obtained,  as  it  is  not  infrequently  needed  in  utilizing  broken  or 
imperfect  observations.  These  can  best  be  obtained  from  observations  on 
slow  moving  stars,  but  any  stars  may  be  used  for  the  purpose.  The  inter- 
vals as  observed,  are  reduced  to  the  equator  by  multiplying  them  by  the 
cosine  of  the  declination  of  the  star  observed. 

The  object  of  these  observations  is  specifically  the  determination  of 
the  error  of  the  chronometer.  This  error  equals  the  right  ascension  of  a 
star  minus  its  observed  time  of  transit,  corrected  for  certain  instrumental 
errors.     These  errors  are  as  follows: 

CORRECTION  FOR  ERROR  OF  LEVEL. 

The  level  error,  designated  by  h,  is  ascertained  from  the  readings  of 
the  striding  level.  The  value  of  a  division  of  the  level  in  seconds  of  time 
must  have  been  previously  ascertained  by  means  of  a  level  trier.  The 
effect  of  the  level  error  is  greatest  at  the  zenith  and  diminishes  to  zero  at 
the  horizon.  The  correction  hi  seconds  of  time  is  given  by  the  following 
equation: 

Coring  cos  (9-f^)  sec  f5  — ?)B. 

When  the  declination  is  north,  it  is  to  be  regarded  as  having  a  plus 
sign  for  upper  and  a  minus  sign  for  lower  culmination.  When  south  it  is 
negative. 


30  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

CORRECTION   FOR   INEQUALITY   OF   PIVOTS. 

This  correction  can  be  made  a  part  of  the  level  correction. 

Let  p  =  tlie  inequality  of  pivots. 

B  =  iuclinatiou  of  axis  giveu  by  level  for  clamii  west. 

B'=  inclination  of  axis  given  bj'  level  for  clamp  east. 

h  =  true  inclination  of  axis  for  clamp  west. 

h'=  true  inclination  of  axis  for  clamp  east. 

,,  B'-B 

then  p— 

4 

h  =  'B  +  J)  for  clamp  west. 

h'  =  W  —  p  for  clamp  east. 

(Gliauvenet,  vol.  ii,  p.  155.) 

CORRECTION   FOR  ERROR    OF   COLLIMATION. 

This  correction,  designated  \>y  c,  is  the  departure  of  the  mean  of  the 
tkreads  from  the  optical  axis  of  the  telescope.  For  stars  at  upper  culmina- 
tion with  clamp  west  it  is  plus  when  the  mean  of  the  threads  is  east  of  the 
axis,  and  minus  when  it  is  west  of  it.  For  stars  at  lower  culmination  the 
reverse  is  the  case.  The  value  of  c  is  one-half  the  difference  between  the 
clock  error  indicated  by  stars  observed  before  and  after  reversal  of  the 
instrument,  divided  by  the  mean  secant  of  the  declinations  of  the  stars. 
This  is  slightly  complicated  with  the  azimuth,  although  the  effect  of  that  is 
largely  eliminated  by  the  proper  selection  of  stars.  Consequently  it  is  to 
be  obtained  by  approximations,  in  conjunction  with  the  azimuth  errors. 
The  correotion  to  be  applied  to  each  star  equals  c  sec  S  zz  cC,  which  is  plus 
for  a  star  at  upper  culmination  and  minus  for  a  star  at  lower  culmination. 
It  is  least  for  equatorial  stars  and  increases  with  the  secant  of  the  declination. 

CORRECTION   FOR   DEVIATION   IN   AZIMUTH. 

This  coiTection,  designated  by  a,  represents  the  error  in  the  setting  of  the 
instrument  in  the  meridian.  Its  effect  is  zero  at  the  zenith  and  increases  toward 
the  horizon.  Since  the  instrument  is  liable  to  be  disturbed  during  the  oper- 
ation of  reversal,  it  is  necessary  to  determine  the  azimuth  error,  both  before 
and  after  reversal,  separately.  A  comparison  of  the  clock  error,  determined 
from  observations  upon  north  and  south  stars,  will  furnish  the  data  neces- 


OBSERVATIONS  FOE  TIME.  31 

saiy  for  the  determinatiou  of  azimuth.  Practically,  it  is  determined  by 
elimination  from  equations  involving  the  mean  of  all  these  stars  observed 
in  each  of  the  two  positions  of  the  instrument,  after  coiTecting-  for  level,  and 
as  it  is  slightly  complicated  with  coUimation  it  must  be  reached  by  two  or 
more  approximations.  The  eiTor  is  essentially  positive  when  the  telescope 
points  east  of  south,  and  negative  when  west  of  south.  The  correction 
applicable  to  any  star  is  expressed  in  the  following  equation: 

Cor.  — «  sin  (9  —  d)  sec  S=zaA. 

It  must  be  understood  that  the  declination  when  north  is  positive  for 
upper  and  negative  for  lower  culmination,  and  that  with  south  declination 
it  is  negative. 

COERECTION  FOE  DIURNAL  ABERRATION. 

The  right  ascension  of  stars,  as  taken  from  the  Berliner  Jahrbuch,  must 
be  corrected  for  diurnal  aberration,  which  equals  0'.021  cos  q)  sec  S.  This 
correction  is  positive  for  upper  and  negative  for  lower  culmination. 

These  corrections  are  suunnarized  in  the  following  equation: 

J  t—a—  (^+aA+&B+cC). 

A,  B,  C,  as  seen  above,  are  constants,  depending  upon  the  latitude  of 
the  place  of  observation  and  the  declination  of  the  star.  Tables  for  these 
quantities  will  be  found  in  an  appendix  to  Annual  Report  U.  S.  Coast  and 
Geodetic  Survey  for  1874. 

The  following  is  an  example  of  the  form  for  record  of  observation  and 
reduction  of  time  observations,  taken  from  a  campaign  for  the  detennination 
of  position  of  Rapid,  South  Dakota. 


32 


A  MANUAL  OF  TOPOGKAPHIC  METHODS. 


Time  determination:  Example  of  record. 


[Kapid      South  Dakota,  November  20,   1890.     Fauth    transit,  No.  534.     Sidereal   chronometer:  Bond  Si,  Sons,  No.  187 
1  divi.sion  ol'  level  =  0"  .118.     Hourlyrate  of  chronometer  =  0". 133.] 


y  Cephei. 

* 

Pegasi. 

u  Pisciura. 

33  Piscium. 

a  Androm. 

Cl-.m 

W. 

W. 

W. 

W. 

■w. 

W. 

Level .; 

Difference  = 

telescope  north 
TT.  Sum.    E. 
d          d      d 
ID.S  -88.1  68.3 
68.2  +87.6  19.4 

-  0.5 

telescope  south. 

W.    Sum.   E. 

d          d       d 
68. 0  +87. 1  19  1 
20.  2  —89.  2  69.  0 

—  2.1 

telescope  south. 

TT.   Sum.    E. 

Add 

20.  0  —89.  5  69.  5 

68.8  +87.2  18.4 

—  2.3 

telescope  south,  telescope  south. 

W.    Sum.    E.     W.   Sxtm.    E. 

d          d       d      d          d       d 

68.  2  +86.  9  18.  7  19.  8  —89.  3  69.  5 

19.  9  —89. 4  69.  5  68.  3  +86.  8  18.  5 

—  2.  5                   —  2.  5 

telescope  north. 

W.    Sum.    E. 

d  .        d       d 

19.  7  —89.  5  69.  8 

68.8  +87.3  18.5 

-2.2 

h. 
23 

23 
6B 

23 

23 

m.      «. 

34  52. 25 

35  11.40 
29.41 
46.78 

36  05.00 

=  4.84 

35     28.97 
-.07 
—.22 
+.(15 

35     28. 83 
34     53. 13 

h. 
23 

23 
23 

47    24.00 
28.55 
32.72 
36.75 
41.09 

3.11 

47    32.62 
—.02 
-.06 
+.03 

47    32. 57 
40    55.67 

h.     m.     s. 

23    54    10. 89 
14.88 
19.22 
23.14 
27.20 

5.33 

23     54    19.07 
—.02 
—.05 
+  .01 

23    54    19.01 
53    41.98 

—37.03 

h. 
00 

00 

00 
23 

m.    s. 

00  13.33 
17.96 
21.94 
25.95 
29.  83 

9.01 

00    21.80 
—.02 
—.04 
+.00 

00    21.74 
59    44.61 

-37. 13 

h.     TO.     ». 

00    03    12.00 
16.83 
21.32 
26.00 
30.85 

7.00 

00    03    21.40 
—.02 
—.06 
+  .00 

00    03    21.32 
00    02    44.42 

rn 

rv 

V 

Correction  for  level 

Correction  for  rate 

R'ednced  transit 

Tabular  E.  A 

—35.  70 

—36.  90 

—36.90 

Mean  of  levels  =  —  2^  ^    /{g  _  _  pggg  ^ ;,      inequality  of  pivots . .  =  .  00 


y  Pegasi. 

Br.  6. 

1  Ceti. 

44  Piscium. 

12  Ceti. 

E. 

E. 

E. 

E. 

E. 

Level •! 

Telescope  south. 

IT.   Sum..    E. 
19.  2  —88.  3  69. 1 
68.9  +87.8  18.9 

d 

Telescope  south. 

W.  Sum.    E. 
68.7  +87.3  18.6 
19. 4  —88.  7  69.  3 

d 

Telescope  south. 

W.  Sum.     E. 
19.  2  —88.  4  69,  2 
68.5  +86.7  18.2 

d 
1.7 

Telescope  north. 

IF.  Sum.    E. 
68.  9  +87.  8  18.  9 
18.9—87.9  69.0 

d 
0.1 

h.    m.      s. 
00    08    05.25 
09.30 

h.    m.      s. 
00    10    05. 00 
22.81 

00    14'   20.70 
24.68 
28.52 
32.90 
37.23 

4.03 

ft.    m.      s. 

00    20     17.35 
20.84 
24.93 
29.16 
33.42 

5.70 

ft. 

00 

TO. 

24 
25 

«. 

56.85 
00.73 
05.37 
09.15 
13.07 

5.17 

lY 

1                 III 

13.  54  1                        39.  30 

II     

17.65 
22.  00 

Sum.=          7. 74 

66.  90 
11     15.49 

- .       9.  50 

I 

00    08    13.55 
—.02 
—.02 
—.02 

00     08    13.49 
00    07     36.59 

39.90 
—.06 
—.09 
—.02 

00     10    39.73 
10    03.56 

28.81 
—.02 
—.02 
—.03 

00    14    28.74 
00    13    51. 75 

25.14 
—.02 
—.02 
—.04 

00     20    25.06 
00     19    48.17 

00 
00 

25 
24 

05.03 
—.02 
—.02 
—.05 

04.94 
27.91 

Correction  for  aberration 

Correction  for  level 6  B= 

Tabular  iJ.  A a= 

a-t= 

—36.  90 

—36. 17 

—36.  99 

—36.  89 

-37.  03 

Div. 
Mean  of  levels  =^  — -^^^ 


'-^^  X  .  118      =  —  .027  =  6.    Ineciuallty  of  pivots  =  .  00. 


LOFGITUDE  DETEEMINATION. 


33 


I++++ 


,1      INN 


+        +    ;++ 


4   i  +  i"  f  I 


I       -    I      :  I  I 


+  I +++    +  1  + 


^< 


MON   XXII 3 


■Jiriir- 


:ii^   <i 


.^3   «  +  + 

"Tip  O  »■« 

II    II      I  ".  § 

5""  +  + 

^    :    a'  ffl  © 

Is   E  +  + 

<i<   -  i.  :- 

I      I  l' 


I    I 


I    "+TS 


3.     i     (r._ 

5-       r°  I    + 


34 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


COMPARISON   OF    TIME. 


After  time  has  beeii  thus  observed  the  chronometers  at  the  two  stations 
shouki  be  compared  by  telegraph. 

Chronometei's  are  compared  in  the  foHowing  manner:  The  chronometer 
at  one  station  being  in  circuit  with  the  chronograph-and  recording  upon  it, 
the  cln-onometer  at  tlie  other  station  is  switched  into  the  general  telegraphic 
circuit,  by  which  it  is  brought  to  the  first  station  and  switched  into  the 
local  circuit  there,  so  that  the  two  chronometers  register  upon  the  same 
chronograph,  their  beats  being  marked  side  by  side  by   the  same  pen. 


Fig.  3.— Switcliboarcl. 


After  this  has  gone  on  for  a  minute  or  more  the  operation  is  reversed,  the 
chronometer  at  the  first  station  is  s\?itched  into  the  telegraphic  circuit  and 
made  to  record  upon  the  chronograph  with  the  clironometer  at  the  second 
station.  Of  course  the  observers  are  informed  of  the  hour  and  minute  at 
which  the  joint  record  upon  the  several  chronographs  begins. 

This  method  constitutes  what  is  known  as  the  automatic  exchange  of 
signals. 

The  arbitrary  exchange  of  signals  is  made  as  follows : 

Each  chronometer  recording  on  its  own  chronograph  as  usual,  and  each 

local  circuit  being  connected  with  the  main-line  circuit,  the  observer  at  one 

station  breaks  the  circuit  by  means  of  the  main-line  talking-key,  which 

break  is  recorded  on  the  chronograph  sheets  at  both  stations.     The  breaks 


COMPARISON  OF  TIME.  35 

are  repeated  at  every  two  secouds  for  at  least  one  full  minute.  The  opera- 
tion is  then  reversed  by  the  observer  at  the  second  station  making  the 
breaks  which  are  recorded  at  both  stations  as  before. 

The  differences  of  time  between  the  chronometers  at  the  two  stations 
are  read  from  the  chronograph  sheets  at  each  station  and  corrected  for 
error  of  the  chronometers.  The  results  from  the  two  chronograph  sheets 
will  differ  by  an  amount  equal  to  twice  the  time  occupied  in  transmission 
of  signals.  The  mean  of  the  two  is  therefore  the  approximate  difference  of 
longitude. 

This  residt  is  yet  to  be  corrected  for  personal  equation,  or  the  differ- 
ence between  the  errors  of  observing  of  the  two  observers.  Every  observer 
has  the  habit  of  recording  a  transit  a  little  too  early  or  too  late,  the  differ- 
ence between  two  observers  not  infrequently  being  as  great  as  a  fourth  of 
a  second.  To  measure  this  difference,  the  observers  usually  meet,  prefera- 
bly at  the  known  station,  both  before  and  after  the  campaign,  and  observe 
for  time  each  with  his  own  instrument,  or  with  one  similar  in  all  respects 
to  that  used  in  the  campaign.  A  comparison  of  the  time .  determinations 
made  by  the  two  observers  gives  an  approximation  to  the  personal  equation. 

A  better  method,  but  one  not  always  practicable,  is  for  the  observers, 
having  completed  half  of  the  observations  for  time  and  longitude,  to  ex- 
change stations  for  the  remainder  of  the  work.  The  mean  of  the  results 
before  and  after  exchange  of  stations  will  eliminate  personal  equation. 

There  is  one  error  incident  to  this  work  which  can  not  be  eliminated. 
This  is  the  unequal  attraction  of  gravity,  or  local  attraction,  or,  as  it  is 
sometimes  called,  station  error.  The  neighborhood  of  a  mountain  mass 
will  attract  the  plumb  line  and  deflect  the  spirit  level  to  such  an  extent  as 
to  cause  serious  errors  in  astronomical  determinations  of  latitude  and  time. 
The  same  restilt  is  frequently  produced  by  a  difference  in  density  of  the 
underlying  strata  of  rock,  so  that  station  errors  of  magnitude  often  appear 
where  they  are  not  expected.  Indeed,  the  station  error  can  not  be  pre- 
dicted Avith  any  certainty,  either  as  to  amount  or  even  direction. 

The  only  practical  method  of  even  partially  ehminating  this  error  is 
to  select  a  number  of  stations  for  astronomical  location,  luider  conditions  as 
widely  diverse  as  possible,  connect  them  by  triangulatiou,  and  by  this 


36  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

means  reduce  all  these  astronomical  determinations  to  one  point,  thus  ob- 
taining for  this  point  a  number  of  astronomical  determinations  each  having 
a  different  station  error.  The  mean  of  these  gives  for  this  point  a  position 
from  which — in  part,  at  least — station  error  has  been  eliminated,  and  this 
mean  position  can  be  transferred  back  by  means  of  the  triangulation  to  the 
several  astronomical  stations,  thus  giving  each  of  them  a  position  similarly 
comparatively  free  from  station  eiTor. 

OBSEEVATIONS    FOR   AZIMUTH. 

The  initial  direction  from  which  the  directions  of  other  lines  in  primary 
triangulation  and  in  primary  traversing  are  computed  is  obtained  by  means 
of  astronomic  observations.  Such  observations  should  be  taken  not  only 
upon  the  initial  line,  but  at  intervals  throughout  the  work  for  its  verification. 
Such  intervals  should  not  exceed  in  the  primary  triangulation  100  miles,  and 
in  primary  traversing  10  to  20  miles. 

Azimuth  observations  are  made  with  the  theodolite  used  in  primary 
triangulation  or  traverse.  The  observations  consist  in  the  measurement  of 
the  horizontal  angle  between  some  close  circumpolar  star,  visually  Polaris, 
and  a  terrestrial  mark,  generally  a  bull's-eye  lantern  set  at  a  distance  of  half 
a  mile  to  a  mile  from  the  observing  station.  The  time  of  observation  on  the 
star  should  be  noted  by  a  chronometer  or  a  good  watch.  As  the  star  is  at 
a  much  higher  angle  of  elevation  than  the  lamp  it  is  necessary  not  only  to 
level  the  instrument  carefully  but  to  measure  the  error  of  level  and  to  cor- 
rect for  it.  It  is  therefore  essential  that  the  value  of  a  division  of  the  level 
bulb  be  known.  These  observations  for  azimuth  may  be  made  at  any  time 
of  the  night,  btit  preferably  they  should  be  made  at  or  near  the  time  of 
elongation  of  the  star,  as  it  is  then  moving  most  slowly  in  aziinuth,  and  any 
eiTor  in  the  time  of  observation  has  the  least  effect  upon  the  resulting  azimuth. 
If  such  observations  be  taken  at  elongation,  no  record  of  time  need  be  made, 
and  the  reduction  of  the  observations  is  simplified.  When  such  observations 
are  made  at  any  other  time  than  at  elongation,  the  time  must  be  noted,  as 
it  forms  an  element  in  the  reduction.  The  error  of  the  clock  or  watch  used 
may  be  obtained  by  comparison  with  railroad  time,  and  corrected  for  the 
difference  in  longitude  between  the  station  and  the  meridian  of  the  railroad 
time.     A  form  of  observation  and  record  is  appended. 


OBSEUVATIOJSrS  FOR  AZIMUTH. 


37 


AZIMUTH    OBSERVATIONS. 


Exain2)le  of  record. 


[Station:  West  base, 


Object. 

Time  P.  M. 

Level. 

3tIicrometer. 

Mean 

. 

1 
Angle. 

West 
end. 

East 
end. 

A.                          B. 

h. 
11 

00 

18 

Div. 
13.9 
50.5 

Div. 
47.] 
10.2 

346 

101 
101 
345 

211 

327 
327 
211 

Telesco] 

'     Div. 
00    14. 8 

32    18.1 
32     19. 8 
58    22.0 

Teiescop 
28    29.0 

05     06.7 
04     26.3 
27     10. 7 

K  direct. 
165    58 

281    31 
281     31 
165    .57 

'.  reverse. 
31     27 

147     03 
147     03 
31     26 

Div. 
25.1 

21.8 
19.7 
01.4 

23.4 

09.5 
00.6 
07.4 

345 

101 
101 
345 

211 

327 
327 
211 

59 

32 
32 
57 

28 

04 
03 
26 

39.9 

09.9 
09.5 
53.4 

22.4 

16,2 
56.9 
48.1 

1 

1     o 

i.115 

1 

J 

^115 

Ills 

|ll5 

32    30.0 

34  16.1 

35  53. 8 
37     08.8 

64.4     1     57.3 
+  7.1 

TT"      t  liatip  Innrvi 

11 
11 

09 
17 

20 
14 

50.  4         10.  3 
13. 8         46. 5 

64.  2         50.  8 
+7.4 

50.  5         10. 1 
12.  9         46  6 

63. 4          56. 7 
+6.7 

11 

26 

22 

14. 3     1     46. 3 
50.1          10.5 

i 
1 

64. 4     I     50. 8 
+7.6 

AZIMUTH    OBSERVATIONS. 


Summary  of  results. 
[Station :  West  base,  Arkansas.    December  27, 1888.] 


Individual  results. 

Combined  results. 

2«*  10   ti^s'^sD. 

1 

I                     38. 80 

«;«;-42.35E. 

J 

55'  t  \  41. 10  R. 
46.3  3 

^                     39.38 

[                     «:||37.65D. 

J 

^|*|43.90D. 

I                     38. 75 

26;  *  I  33.  60  E. 

J 

«:;^47.05E. 

V                     40.10 

«:^^33.15D. 

) 

Grand  mean 

294    10    39.26 

38  A  MANUAL  OF  TOPOGEAPHIG  METHODS. 


REDUCTION    OF   AZIMUTH    OBSERVATIONS. 

The  time  of  observation  of  a  star  is  first  to  be  corrected  for  the  differ- 
ence in  longitude,  assuming  that  railroad  time  has  been  used,  and  for  the 
error  of  the  watch.  It  is  then  reduced  from  mean  to  sidereal  time.  From 
the  sidereal  time  of  observation  is  to  be  subtracted  the  right  ascension  of 
Polaris,  if  that  star  is  used,  which  is  given  in  the  Nautical  Almanac,  the 
result  being  the  hour  angle  or  the  sidereal  time  which  has  elapsed  since  it 
passed  the  meridian  of  the  place  of  observation,  given  in  hours,  minutes, 
and  seconds.  This  result  is  to  be  converted  into  degrees,  minutes,  and 
seconds. 

m        i        \  «  sill  t 

Then  tan  A 


1-b  cos  t 
where  a— sec  9  cot  6,      y=the  latitude. 


h  =  '^ 


tan  (J) 


tan  6  (5— the  declination  of  star. 

t-z  hour  angle. 
A  zz  angle  between  north  pole  and  the  mark. 

This  angle  is  to  be  con-ected  for  level  as  follows: 

level  corr.— —  A$(w  +  ^t;')— (e  +  e')Uau.7i. 

d  being  the  value  of  a  division  of  the  level. 
■w-}-'w',  readings  of  west  end  of  level  bubble. 
e+e',  readings  of  east  end  of  level  bubble. 
h,  the  angular  elevation  of  pole  star. 


AZIMUTH  OBSERVATIONS.  '  39 

An  example  of  reduction  is  as  follows: 

AZIMUTH    OBSERVATIONS. 

Example  of   reiluciion. 

[Station:  West  base;  December  27,  1888.    Observer,  S.  S.  G.    Latitude=34    45    20.8    Longitude  92    13    31.5.] 

h.  VI.  .s. 

Tinie  of  observation  =  Tw                                    =  11  00  18 

Correction:  ninetieth  meridian  time  to 92^.215  =  —  8  54 

"Watch  slow;  ninetieth  meridian  time  +  02  ^ 

local  mean  time  * 

Correction ;  mean  to  sidereal  time 
Kight  ascension  mean  sun 


Tm=:10 
~18 

51    2B 
+  1     47 
20    36 

=  29 
—    1 

19    49 
18    25 

t      •      =28 
—  24 

01     24 

h. 
t  (time)  =    4 

t  (arc)     =60 

01     24 
21     00 

ec  }1  cot  6    b  = 

_  tan  0 
tan  S 

log  b 


logtanA  178    38    08.0    =8.3769185              0.9923296         =1—6  cost 

angleto  +115'  32    30.0 

mark  , 

Level  corr.  -3-8    le^el  corr.  = -|l  j  (^^.,„,)_,g_g,,  |  t^^  ,, 

o       ,         „  3"]         Div.  ^      •" 

Az.  ofmai-k  =  294     10    34.2  = — j~    X  7. 1  X  .  694  =  — 3.8 

When  observations  for  azimuth  are  to  be  made  at  elongation,  it  is  nec- 
essarv  to  know  the  mean  time  of  elongation.  This  is  computed  by  the 
following  method:  the  hour  angle  at  elongation  is  obtained  from  the  follow- 
ing equation: 

cos  te  =tan  <p  cot  S. 

The  hour  angle  plus  the  right  ascension  of  the  star  gives  the  sidereal 
time  of  its  western  elongation,  which,  reduced  to  mean  time,  gives  the  local 
mean  time  in  question. 

The  azimuth  of  a  pole  star  at  elongation  is  determined  by  the  use  of 

the  following  equation : 

sin  A  zz  sec  cp  cos  S. 


40  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

The  following  is  an  example  of  these  computations: 

Example  of  the  compntation  of  the  azbmith  at  elongation^  and  the  local  mean  times  of  both 
elo7igations  of  Polaris. 

[Latitude  =  (^ —40°.    Meridian  of  "NVasbingtou.    November  28, 1891.] 


•    Sine  Azimntli 
log.  sec.  40° 

at  elongation  = 

=  sec.  (J  cos  S. 
=  0. 1157460 

log.  cos  ij 

88    44    05.5 
1     39     05.  S 

=  8. 3439803 

log.  sine  A 

=  8. 4597263 

Cos.  hour  angle  at  elongation,  t 
log.  tan  40° 

e.   =tan  <f>  cot  6. 
=  9.9238135 

log.  cot  6 

88    44    05.5 
88    56    17.5 

=  8.3440862 

log,  cos  te 

=  8. 2678997 

t«  =  5    55    45.  2. 
Sidereal  time  western  elongation,  Ta  :=;  E.  A.  Polaris  +  te. 


=  1 

19 

35.2 

=   5 

56 

45.2 

T.=   t" 

15 

20.4 

o,=16 

29 

14.4 

=  9 

13 

54.0 

1 

30.7 

9 

12 

23. 3     Not.  28. 

=    2 

47 

36.7     A.M.,  Nov.  28. 

h. 

71!.        s. 

a  —  te  = 

:19 

23    50.0 

as  ^ 

16 

20     14. 4 

^ 

:     2 

54     35.6 

= 

0     28.6 

Sidereal  time  eastern  elongation  =  24'  +  a 


For  longitudes  west  of  Wasliiugtou  decrease  times  of  elongation  0.66  for  each  degree. 


CHAPTER     III. 


HORIZONTAL  LOCATION 


The  primary  control  or  geometric  work  is,  in  the  ordinary  case,  effected 
by  tiiangulation.  Wherever  this  is  not  practicable  or  not  economic,  resort 
is  had  to  what  is  known  as  primary  traversing,  but  wherever  the  country  pre- 
sents sufficient  relief  for  the  purpose,  triangulation  is  employed,  as  it  is  more 
accurate  and  cheaper.  In  some  parts  of  the  country  triangulation  of  suffi- 
ciently accurate  character  for  controlling  the  map  has  been  executed  by 
other  organizations,  notably  by  the  U.  S.  Coast  and  Geodetic  Survey,  and  the 
U.  S.  Lake  Survey.  Wherever  such  triangulation  is  available,  the  results 
should  be  adopted  and  utilized  for  the  control  of  the  maps. 


PARTY  ORGANIZATION. 


The  primary  triangulation  is  generally  carried  on  by  a  special  party. 
It  is,  however,  on  some  accounts  and  under  certain  circumstances,  economi- 
cal and  advisable  that  all  the  work  be  done  by  one  and  the  same  party. 
The  disadvantage  is  that  it  divides  the  time  and  attention  of  the  topographer, 
requiring  him  to  turn  his  attention  from  one  thing  to  another;  the  advan- 
tage, that  it  insures  the  selection  of  such  points  as  are  needed  by  the 
topographer  for  carrying  forward  the  work.  If  the  work  is  done  by  a  special 
party,  the  points  selected  are  more  likely  to  be  chosen  on  account  of  their 
forming  good  figures  in  the  triangulation,  than  on  account  of  their  conve- 
nience and  usefulness  to  the  topographer.  The  secondary  triangulation,  the 
traversing,  and  the  sketching  are  usually  carried  on  by  different  men,  but 
under  a  single  party  organization.  The  sketching  is  done  by  the  chief  of 
party,  the  secondary  triangulation  and  height  measurement  l)y  his  most 
experienced  assistant,  while  the  traversing,  with  height  measurement,  is  done 
by  the  other  assistants. 


42  A  MANUAL  OF  TOPOGEAPHIG  METHODS. 

BASE-LINE    MEASUREMENT. 

This  is,  ordinarily,  tlie  first  of  tlie  preparatory  steps  toward  map  making. 
Upon  the  proper  selection  of  the  site  of  the  base  line  and  its  correct  meas- 
m-ement  depends  all  the  subsequent  work  of  tri angulation.  The  site  must 
be  reasonably  level.  It  is  not  essential  that  it  be  absolutely  so,  but  the 
more  closely  it  approaches  a  plane  the  less  difficulty  will  be  experienced  in 
making  an  accurate  measurement.  The  site  should  afford  sufficient  room 
for  the  measurement  of  a  base  from  5  to  10  miles  in  length.  A  base  less 
than  5  miles  in  length  is  not  an  economical  one,  inasmuch  as  it  is  less 
costly  to  extend  the  base  than  to  complicate  the  expansion.  A  greater 
length  than  10  miles  is  imnecessary,  because  this  length  permits  of  easy 
expansion,  and,  if  the  length  be  greater  than  this,  it  may  be  difficult  to  con- 
struct signals  at  the  two  ends  of  the  base,  which  will  be  intervisible. 

The  ends  of  the  base  must  be  intervisible,  and  they  must  be  so  situated 
with  regard  to  suitable  points  for  expansion  and  triangulation  as  to  form 
well  proportioned  figui-es.  Whenever  possible,  the  base  line  should  form  a 
side  or  diagonal  of  a  closed  quadiilateral  or  pentagonal  figm-e. 

While  it  is  unnecessary  to  devote  time  to  obtaining  the  extreme  of 
accuracy  in  the  measiu-ement  of  a  base,  this  measurement  should  be  so 
acciu-ate  that  its  errors  can  not  affect  the  map,  although  multiplied  many 
times  in  the  associated  triangulation.  All  necessary  precaution  should  be 
taken  to  secui-e  this  result. 

Various  methods  and  instruments  have  been  employed  in  the  measure- 
ment of  base  lines  upon  the  Geological  Survey.  At  first  wooden  rods  were 
employed,  varnished  and  tipped  with  metal.  When  used  in  measuring, 
these  were  supported  upon  trestles  and  contacts  made  between  them,  with 
considerable  refinement.  The  advantage  of  using  these  rods  consisted  in 
the  fact  that  their  length  is  but  slightly  affected  by  temperatm-e,  which  is 
the  main  source  of  error  in  base-line  measurement,  and  being  thoroughly 
varnished  they  were  not  greatly  affected  by  moistm-e. 

Subsequently  bars  of  metal  were  employed  of  the  pattern  known  as 
the  Coast  Survey  secondary  bars.  These  consist  each  of  a  steel  rod  between 
two  zinc  tubes.  As  the  two  metals  expand  at  different  rates  under  changes 
of  temperature,  their  relative  lengths  at  any  temperature  as  compared  to  the 


BASE  LINE  MEASUEEMENT.  43 

i-elative  lengths  at  a  normal  temperature  is,  theoretically,  an  indication  of  the 
temperature  of  the  bars  at  any  time.  The  arrangement  for  indicating  their 
relative  lengths  forms  a  part  of  the  apparatus,  and  is  intended  to  indicate 
the  temperature  of  the  bars,  and  thus  to  afford  means  of  reducing  the  lengths 
of  the  bars  to  a  normal  temperature.  It  has  not  been  found,  however,  to 
work  well  in  practice.  Besides  this,  there  are  other  objections  to  the  use  of 
bars  of  any  kind,  which  may  be  summarized  as  follows:  First,  their  use  is 
expensive.  A  considerable  number  of  men  are  needed,  and  as  the  measure- 
ment proceeds  slowly  it  often  requires  from  a  month  to  six  weeks  to  measure 
and  remeasure  a  base  five  miles  in  length.  Again,  since  these  bars  are  but 
four  meters  in  length,  there  are  many  contacts  to  be  made  in  each  mile  of 
measurement,  and  each  contact  affords  the  possibility  of  a  trifling  error. 

In  view  of  these  objections  and  of  certain  positive  advantages  which 
the  change  would  produce,  it  was  decided,  in  1887,  to  drop  the  use  of  bars 
in  the  measurement  of  base  lines,  and  to  adopt  in  their  place  long  steel 
tapes.  By  their  use  it  has  been  found  easy  to  attain  the  required  degree  of 
accuracy  in  measurement,  inasmu.ch  as  the  number  of  contacts  is  reduced 
to  a  small  fraction  of  the  number  necessary  in  the  use  of  bars,  while  the 
uncertainty  in  regard  to  the  temperature  of  the  measuring  apparatus  is 
reduced  to  a  minimum  by  carrying  on  the  measurement  at  night  or  in  cloudy 
weather.  The  expense  of  the  measurement  is  greatly  reduced.  Fewer 
men  are  required.  The  work  of  preparing  the  ground  and  the  work  of 
measuring  are  much  lessened,  and  the  rapidity  of  measuring  is  increased 
manyfold.  The  diminished  cost  makes  it  practicable  to  measiire  much 
longer  bases,  thus  diminishing  the  number  -of  stations  required  in  the 
expansion.  It  allows,  also,  a  measurement  of  base  lines  at  shorter  intervals 
in  the  triangulation. 

The  tape  in  use  has  a  length  of  300  feet.  It  should  be  carefully  com- 
pared, at  an  observed  temperature,  with  the  standard  of  the  U.  S.  Coast  and 
Geodetic  Survey,  both  before  and  after  its  use  in  base  measurement.  Prefer- 
ably, the  site  for  the  base  line  should  be  selected  along  a  railway  tangent, 
as  such  a  location  is  approximately  level,  and  the  railway  ties  afford  an 
excellent  support  for  the  tape.  If  such  a  location  can  not  be  obtained,  it 
should  be  selected  so  as  to  till  the  requirements  above  mentioned,  cleared 


44  A  MANUAL  OF  TOPOGliAPHlO  METHODS. 

of  brush  and  undergrowth,  and,  if  necessary,  its  sharp  inequalities  should 
be  leveled.  The  tape  should  be  supported  by  a  series  of  low  stools,  whose 
legs  are  pressed  into  the  ground  at  intervals  of  not  more  than  25  feet,  while 
similar  stools  should  sustain  each  end  of  the  tape. 

The  personnel  required  in  the  measurement  of  a  base  line  is,  in  an 
ordinary  case,  as  follows: 

First.  The  chief  of  the  party,  who  exercises  a  general  supervision  over 
the  work,  marks  the  extremities  of  the  tape  and  provides  the  necessary  pre- 
cautions against  errors  in  the  measurement,  as  hereafter  stated. 

Second.  The  rear  chainman,  who  adjusts  the  rear  end  of  the  tape  to  the 
contact  marks  and  who  carries  and  reads  one  of  the  thermometers. 

Third.'  The  head  chainman,  who  adjusts  the  forward  end  of  the  tape? 
exerts  the  requisite  tension  upon  it,  and  carries  and  reads  a  second  ther- 
mometer. 

Fourth.  A  recorder. 

The  measurement  of  a  base  with  the  steel  tape  is  a  simple  matter. 
Provision  must,  however,  be  made,  first,  for  the  proper  alignment  of  the 
base ;  second,  for  the  proper  tension  of  the  tape ;  and,  third,  for  the  measure- 
ment of  temperature. 

The  alignment  is  a  simple  matter,  and  is  generally  marked  out  upon 
the  gi'ound  in  advance  of  the  work  of  measurement.  In  cases  where  a 
railway  tangent  furnishes  the  site  for  the  base  line,  no  alignment  is  needed 
beyond  the  provision  for  keeping  the  tape  always  at  a  uniform  distance 
from  one  of  the  rails. 

For  insuring  a  uniform  tension  of  the  tape,  an  ordinary  spring  balance 
is  used,  which  is  attached  to  the  forward  end  of  the  tape,  where  a  tension 
of  twenty  pounds  is  applied.  In  order  to  apply  this  uniformly,  and  to 
insure  against  «lip  of  the  tape,  an  apparatus  de\'ised  by  Mr.  H.  L.  Baldwin, 
jr.,  of  the  Geological  Survey,  is  in  use. 

For  its  use,  it  is  necessary  to  obtain  strips  of  board  about  five  feet  long 
and  four  inches  in  width,  in  number  equal  to  the  number  of  lengths  of  tape 
of  which  the  base  line  consists.  Numbered  strips  of  zinc  of  equal  nmnber, 
each  about  eight  inches  long  and  an  inch  in  width,  are  tacked  to  blocks  of 
wood,  and  these  blocks  of  wood  in  turn  nailed  down  upon  the  boards  above 


BASE  LINE  MEASUEEMENT.  45 

ineiitioned,  while  the  boards  are,  m  case  measurement  is  made  along  the 
railway  tangent,  nailed  down  to  the  railway  ties.  These  boards  are 
designed  to  support  the  devices  for  maintaining  the  tension,  and  the  con- 
tacts are  marked  upon  the  strips  of  zinc.  Mr.  Baldwin's  apparatus  consists 
essentially  of  a  wheel  worked  by  a  lever  and  held  by  ratchets  in  any 
desired  position.  This  wheel  is  attached  to  the  spring  balance  in  such  a 
way  that  by  turning  it  the  strain  is  put  uj)on  the  spring  balance,  which  is 
held  at  the  desired  tension  by  the  ratchets.  A  small  mechanism  at  the  rear 
end  of  the  tape  is  employed  to  hold  the  zero  of  the  tape  at  the  opposite 
mark.  The  great  length  of  the  tape,  300  feet,  allows  considerable  friction 
or  drag  when  the  supports  are  frequent,  and  in  order  to  insure  a  reasonably 
uniform  distribution  of  the  strain  upon  the  tape,  it  should  be  raised  and 
allowed  to  fall  with  the  strain  on. 

The  measurements  should  be  made  at  night,  or  during  cloudy  days, 
in  order  that  the  temperature  of  the  air,  which  is  that  indicated  by  the 
thermometers,  and  that  of  the  tape  be  as  nearly  as  possible  the  same.  The 
temperature  must  be  carefully  observed  by  at  least  two  thermometers  at 
each  tape  length,  in  order  that  the  best  ]possible  data  for  temperature  cor- 
rection may  be  obtained. 

The  base  should  be  measured  at  least  twice,  and  the  two  results  com- 
pared by  sections  of  1,200  feet,  or  four  tape  lengths.  The  ends  of  the 
base  must,  if  possible,  be  permanently  marked  by  means  of  stone  monu- 
ments set  into  the  ground  so  that  their  surfaces  are  but  a  few  inches  above 
its  level  and  the  exact  position  of  the  ends  should  be  indicated  by  a  cross 
cut  in  a  copper  bolt  embedded  in  the  head  of  a  stone,  in  order  that  the 
base  may  be  preserved  for  futm-e  references. 

A  line  of  levels  must  be  run  over  the  site  or  over  the  stools  which 
support  the  tape  for  the  purpose  of  obtaining  its  profile  and  thereby  the 
means  for  deducing  its  horizontal  length. 

REDUCTION  OF  BASE  LINE  MEASUREMENT. 

The  first  correction  to  be  applied  is  that  of  redustion  to  a  standard. 
The  correction  for  this  is  obtained  by  comparison  with  the  standard  of  the 
U.  S.  Coast  and  Geodetic  Survey.     The  correction  for  the  entire  line  is  in 


46  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

proportion  to  the  correction  as  obtained  by  comparison  with  the  standard. 
If  the  tape  be  longer  than  the  standard,  the  correction  will  be  positive,  if 
shorter,  negative. 

Second.  The  correction  for  inclination,  the  data  for  Avhich  are  obtained 
by  rnnning  a  line  of  levels  over  the  base  line.  This  line  of  levels  gives  the 
rise  or  fall,  in  feet  and  decimals  of  a  foot,  between  the  points  of  change  in 
inclination.     From  this  and  the  measured  distance  the  angle  of  inclination 

is  computed  from  the  formula,  sin  0  =  p ;  R  being  the  distance  and  h  the 

difference  in  height,  both  given  in  feet.  The  correction  in  feet  to  the  dis- 
tance is  then  computed  by  the  equation,  » 

Corr.zr  ^"f  ^  6''  R  or  0.00000004231  9^  R,  G  being  expressed  in  minutes. 

(See  Lee's  Tables,  p  83.) 

Third.  The  correction  for  temperature.  Steel  expands  for  each  degree 
of  temperature  .0000063596  of  its  length.  This  fraction  multiplied  by  the 
average  number  of  degrees  of  temperature  at  the  time  the  base  line  was 
measured  above  or  below  sixty-two  degrees,  which  is  taken  as  the  normal 
temperature,  gives  the  proportion  in  which  the  base  line  is  to  be  diminished 
or  extended  on  account  of  this  factor.  Care  must  be  taken  to  obtain  cor- 
rectly this  average  temperature.  It  must  be  the  mean  of  all  the  thermo- 
metric  readings,  taken  at  uniform  intervals  of  distance  during  the  measure- 
ment. If  the  temperature  be  above  the  normal,  the  correction  is  positive, 
and  vice  versa. 

Fourth.  The  reduction  to  sea  level.  The  base  line  is  measured  on  a 
cii-cle  parallel  to  the  sea  surface  and  raised  above  it,  at  an  elevation  which 
is  known  at  least  approximately.  This  circle  with  radii  drawn  therefrom  to 
the  center  of  the  earth  forms  approximately  a  triangle  similar  to  that  formed 
by  the  radii  of  the  earth  with  the  sea  surface.  The  length  at  sea  level  is 
derived  with  a  sufficient  approxiination  to  correctness  by  the  proportion: 
R:  h::  K:   correction. 

R  being  the  radius  of  the  earth,  h  the  mean  height  of  the  base  line 
above  sea  level,  and  K  its  measured  length.  (See  Report  U.  S.  Coast  and 
Geodetic  Survey,  1882,  Appendix  9,  p.  196.) 


BASE  LINE  MEASUREMENT. 


47 


The  following-  is  an  example  taken  from  the  records  of  measiu-ement 
in  1889  near  Spearville,  Kansas,  together  \'vith  the  reduction  of  this  base 
for  inclination,  temperature,  and  elevation  al  ove  sea  level: 

Eeeord  of  measurement  and  reduction  of  Spearville  hase,  Kansas. 
[Section  1.    Stations  0-10.  October  16,  1889.    Light  rain  falling.] 


IXo.  of  Tape. 

Time, 

Tension. 

Tliermometera. 

Temperature 
correction. 

,  Totcal  length  of  section. 

A. 

B. 

1 

h.       m. 
10     13 
20 
26 
31 
37 
42 
47 
51 
55 
58 

Founds. 
19.75 
20.00 
20.00 
20.25 
20.00 
20. 125 
20.  25 
20.00 
20. 125 
20.00 

50.5 
50.5 
50.5 
50.5 
50.7 
51.5 
51.0 
50.8 
50.8 
50.7 

50.0 
50.0 
50.0 
50.0 
50.5 
50.6 
50.8 
50.2 
50.  0 
50.5 

iloan  temp.  =  50.51 

62-50.51  =  11.49 

—11.49X3000. 

X  .000006 

=  -.207 

1  tape  length =300. 0617 

10  :■  300.  0617 =  3,  000.  617 

Temperature  corr — .  207 

Eesult  first  measurement=  3, 000. 410 

[Second  measurement,  October  17, 1889.] 


No.  of  Tape. 

Time, 
p.m. 

Tension. 

Thermometers. 

Temperature 
correction. 

Total  length  of  section. 

A. 

?• 

k.         m. 
12    13 
21 
25 
29 
33 
36 
38 
41 
45 
50 

Pounds. 
20.00 
20.25 
20.00 
19.75 
20.00 
20.  00 
20.00 
20. 12 
19.75 
20.13 

52.3 
53.3 
53.8 
55.0 
55.0 
53.8 
54.0 
54.5 
55.1 
54.5 

52.4 
52.9 
54.0 
54.8 
53.2 
54.0 
54.0 
54.0 
54.4 
54.1 

Mean =53. 96 

62     —53.96=8.04 

-  8.  04  X  3000. 

X.  000006 

=  — 145 

Tape  set  hack  from  sta.  0      .85  inch. 
= .  071  foot. 

2      

4. 

Temperature  corr — .  145 

Eesultsecondmea3urement=3,000.40] 

Correction  for  inclination  Sjyearville  base, 
Correction  =5HLiiL:  92  x  Distance. 


Approximate 

distance. 

Differ- 
ence of 
elevation. 

Angle  e 

log  e 

2  log  9 

log 

Sinn' 

2 

log  (list. 

log 
correction. 

Correction. 

Feet. 

Feet. 

,      „ 

200 

0.8 

13    34 

1. 1326 

2. 2652 

2.  6264 

2. 3010 

7. 1926 

.0015 

4,200 

4.2 

2    22 

0.  3674 

0. 7348 

3.6232 

6. 9844 

.0010 

4,000 

12.0 

10    08 

1.  0052 

2.  0104 

3.  6021 

8. 2389 

.0173 

1,000 

1.0 

3    23 

0.  5250 

1.  0501 

3. 0000 

6.  6765 

.0005 

2,000 

3.0 

5    04 

0.7024 

1.4049 

3.3010 

7. 3323 

.0021 

4,200 

22.0 

12    23 

1. 0917 

2.  1834 

3.6232 

8. 4330 

.0271 

2,800 

7.0 

8    27 

0.9263 

1.  8527 

] 

3.4472 

7.  9263 

.0084 

1,000 

0.0 

0    00 

0.0000 

0.  0000 

Constant.    { 

3.  0000 

0. 0000 

.0000 

1,000 

1.0 

3    23 

0.  5250 

1.  0500 

3.0000 

6.  6764 

.0005 

4,200 

20.0 

11     16 

1.  0504 

2. 1008 

3. 6232 

8. 3504 

.  0224 

3,800 

6.0 

5    20 

0. 7267 

1.4535 

3.  5798 

7. 6597 

.0046 

2,000 

4.0 

6    45 

0.  8293 

1.  6dS6 

3.  3010 

7.5860- 

.0038 

5,400 

31.4 

19    39 

1.  2934 

2.  5867 

3.7324 

8. 9455 

.0882 

2,000 

2.6 

4     24 

0.  6-137 

1.  2874 

3.3010 

7.2148 

.0016 

135 

0.05 

1     18 

0. 1072 

0.2144 

2. 1303 

4.  9712 

.0000 
.1790 

48 


A  MANUAL  OF  TOPOGKAPHIO  METHODS. 


Beductioii  to  sea  level. 


Correction 

lo^  K  (meti'cs) . 
log  A  (metres). 
Co  log  R 


Spearville  base :  Summary  by  sections. 
[Corrected  for  temperature.] 


.  =  4.  03956 
.  =  2.  87599 
.   3. 19660 


Stations. 

First 
measure. 

measurl       1     I^'^—- 

1 

1  to    10 
ID         20 
20          30 
30          40 
40          50 
50          60 
60          70 
70          80 
80          90 
90        100 
100        110 
110        119 
119        126 

3, 000.  410 

.418 

.431 

.426 

.437 

.417 

.369 

.306 

.955 

.676 

3,  000.  899 

2,  700.  581 

2, 100.  244 

3,000.401 

.393 

.431 

.446 

.478 

.455 

.392 

.350 

.938 

.667 

3,  000.  898 

2,  700.  571 

2, 100.  234 

First— Second. 

+  .009 
+.025 
+.000 
-.020 
-.041 
-.038 
-.023 
+.010 
+.017 
+.009 
+.0U1 
+.  010 
+.010 

37,806.629 

37,  806.  660 

-.  031=.  372 

Mean  of  2  raea^urement.'i =  *37, 806.  645 

Reduction  from  S.  ^V.  lia.se  to  A -        168. 235 

Reduction  from  N.  E.  base  to  A -  2. 864 

Correction  for  inclination —  0. 179 

Reduction  to  sea  level —  4.448 

Corrected  length =   37,630.919 


PRIMARY  TRIANGULATION. 

The  base  line  having  been  measured,  the  next  step  is  the  expansion. 
This  work,  as  well  as  the  body  of  the  triangulation,  consists  in  the  selection 
of  stations,  the  erection  of  signals,  and  the  measurement  of  angles.  Each 
triangle  proceeding  from  the  base  line  outward  will,  when  the  angular  meas- 
urement is  completed,  have  one  side  and  the  three  angles  known,  from  which 
the  other  two  sides  can  be  computed  by  means  of  a  simple  trigonometric 
fonniila. 

The  expansion  diffei's  from  the  body  of  the  triangulation  only  in  the 
fact  that  the  average  length  of  the  sides  of  the  triangles  is  less.  As  the 
expansion  progresses  away  from  the  base  line,  the  sides  of  successive  triangles 
become  gradually  longer,  until  the  average  length  of  side  of  the  triangula- 
tion is  reached.     Since  the  sides  are  increasing  in  length,  and  hence  since  any 

*  Con'ected  for  temperature. 


PEIMAEY  TRI ANGULATION.  49 

inaccuracy  in  the  measurement  of  the  base  is  multiplied,  this  work  must 
be  planned  and  executed  with  greater  care  than  the  body  of  the  triaugula- 
tion  requires. 

A  base  line  measui-ed  as  above  prescribed  requires  little  expansion, 
since  from  the  extremities  of  an  8  or  10  mile  base  one  can  observe 
directly  on  points  12  to  15  miles  away,  a  distance  as  great  as  the  average 
side  of  a  triangle.  Ordinarily,  from  the  ends  of  the  base,  the  surveyor 
can  observe  directly  upon  stations  in  his  scheme  of  triangulation. 

In  the  western  mountain  region,  where  the  sides  of  triangles  may  be 
20  to  50  miles  in  length,  an  expansion  is  required. 

SELECTION   OF    STATIONS. 

In  the  selection  of  triangulation  stations  two  different  sets  of  require- 
ments must  be  served. 

First.  They  iTiust  be  so  selected  as  to  afford  what  is  known  as  strong 
figures,  in  order  to  reduce  to  a  minimum  the  errors  which  will  creep  into  an 
extended  system.  In  order  to  insm-e  intervisibility,  they  should,  if  possible, 
be  located  upon  hill  or  mountain  summits,  the  most  commanding  in  the 
neighborhood.  No  triangle  upon  which  dependence  is  placed  for  the  loca- 
tion of  a  station  should  have  at  that  station  an  angle  of  less  than  30°  or 
more  than  150°. 

The  stations  should,  if  practicable,  be  grouped  into  simple  figures,  as 
quadrilaterals,  or  pentagons  with  an  interior  station,  etc.  In  cases  where  an 
area  is  being  covered  with  triangulation,  such  groupings  naturally  occur, 
but  in  certain  cases  the  triangulation  takes  the  form  of  narrow  belts  of  fig- 
ures, and  then  the  belt  may  consist  of  simple  triangles  or  quadrilaterals,  as 
more  complex  figures  are  rarely  desirable. 

Second.  Since  the  sole  object  of  this  triangulation  is  the  control  of  the 
topographic  map,  the  location  of  stations  must,  as  far  as  is  consistent  with 
accuracy,  be  adjusted  to  the  needs  of  the  topographers.  This  requirement 
affects  most  seriously  the  distance  between  stations.  Every  atlas  sheet 
must  contain  at  least  two  primary  stations,  and  a  third  is  desirable.  Thus, 
for  controlling  the  sheets  on  the  scale  1 :  62500,  the  stations  should  not  be 
more  than  10  or  12  miles  apart,  and  should  be  located  with  du-ect  reference 

MON   XXII i 


50  A  majstual  of  topographic  methods. 

to  the  control  of  certain  sheets.  Again,  since  the  primary  stations  must  be 
occupied  by  topographers  for  intersecting  on  numerous  points,  they  must 
be  selected  with  reference  to  this  requirement.  They  should  command  an 
extended  view,  especially  of  points  suitable  for  cutting  in,  such  as  hill  and 
mountain  summits,  houses^  churches,  etc. 

The  instrument  should,  wherever  possible,  be  accurately  centered  under 
the  signal.  Whenever  it  is  necessary  to  set  up  off  center,  the  direction  and 
distance  to  the  signal  should  be  carefully  measured  and  recorded. 


While  signals  should  be  of  the  simplest  and  least  expensive  form  which 
will  serve  the  pm-pose,  their  form  and  material  must  depend  upon  the  requhe- 
ments  and  the  materials  at  hand.  In  a  mountainous  country,  where  the 
summits  ai'e  treeless,  simple  cairns  of  stone,  7  to  10  feet  in  height,  are  em- 
ployed. Where  the  summits  are  wooded,  it  is  frequently  convenient  to  clear 
them,  leaving  a  single  tree  to  serve  as  a  signal.  In  such  cases  it  is  advisable 
to  trim  the  tree  of  branches,  with  the  exception  of  a  tuft  at  the  top.  Where 
the  station  is  clear,  but  with  green  timber  easily  accessible,  it  is  advisable 
to  make  a  tripod  of  small  trees,  each  with  a  tuft  at  its  top.  In  undulating 
and  hill  country  it  is  often  necessary  to  erect  scaffolds.  These  should  be 
built  of  sawed  lumber  and  framed  in  simple  fashion.  If  the  lines  are  short, 
a  pole  with  a  flag  may  be  set  in  the  top.  If  the  lines  are  long,  the  tower 
itself  may  serve  as  a  signal,  in  which  case  its  upper  part  should  be  clothed 
in  black  and  white  cotton. 

The  annexed  cut  shows  a  form  of  framed  signals  adapted  for  use  on 
the  treeless  plains  of  Kansas  and  the  rolling  open  hills  of  New  England, 
and  elsewhere,  where  observing  towers  are  not  necessary.      (PI  rv.) 

It  is  frequently  necessary  to  raise  the  instmment  to  a  considerable  ele- 
vation above  the  ground,  in  order  to  overlook  surrounding  obstacles.  In 
such  cases  the  structiu-es  for  supporting  the  instrument  should  be  combined 
with  the  signals,  and  hence  they  may  properly  be  described  and  figured 
here.  These  observing  towers  should  be  in  two  parts.  An  interior  struc- 
ture, solidly  built  of  sawed  lumber,  if  available,  for  the  immediate  support 
of  the  instrament,  and  a  framework  surrounding  it,  supporting  a  platform 


SiaiSTALS. 


51 


just  below  the  staud  for  the  instrument,  for  the  observer.  The  two  should 
be  separate,  in  order  that  the  jan-ing  incident  to  moving  about  on  the  plat- 
form be  not  communicated  to  the  instrument.  Such  a  type  of  obser^dng 
tower  is  figured  in  Fig.  4. 


Fig.  4.— Sigual  and  instrunioiil;  suiipurt. 


When  sawed  lumber  is  not  obtainable,  other  material  must  be  used. 
In  the  Sierra  Nevada  of  California,  among"  the  sugar-pine  forests,  a  support 


52 


A  MANUAL  OF  TOrOGEAPHIC  METHODS. 


for  the  iustrumeut  is  not  unfrequently  obtained  by  sawing  off  the  top  of  a 
high  tree,  and  setting  the  instrument  upon  the  stump,  50  or  75  feet  above 
the  ground,  the  tree  being  guyed  out  by  wire  cables  to  prevent  swaying  in 
the  wind.  The  phxtform  for  the  observer  is  supported  by  neighboring  trees, 
similarly  sawed  off  and  supported  for  the  purpose.  Similar  devices  are 
resorted  to  also  in  the  forests  of  AVest  Vu-ginia,  Kentucky,  and  Tennessee. 
In  the  secondary  triangulation  in  these  regions,  the  instrument  support  is, 
in  many  cases,  provided  as  above  described,  while  the  observer's  platform, 
instead  of  having  an  independent  support,  is  attached  to  the  same  tree.  This 
is  objectionable,  but  is  often  the  best  that  can  be  done. 


Fig.  5. — Coast  Survey  Heliotrope. 

In  other  cases  it  is  more  economical  to  suppoi't  the  instrument  upon  the 
ground,  and  to  have  openings  made  thi'ough  the  forest  upon  the  station  hill, 
in  the  du-ections  of  the  sight  lines,  or  even  to  have  the  whole  summit  cleared. 

It  is  not  infrequently  necessary  to  use  more  elaborate  forms  of  signals, 
especially  when  the  point  observed  upon  is  below  the  horizon  line,  so  that 
the  background,  instead  of  being  the  sky,  consists  of  forests  or  brown  plains. 
In  such  cases  resort  is  had  to  heliotropes.  These  are  simply  instruments  for 
reflecting  the  sunlight  to  the  observer  at  the  instrument.  The  simplest  form 
is  a  circular  mirror  with  a  screw  hinged  at  the  back,  giving  a  universal 
motion.  This  is  screwed  into  a  stake  or  tripod  over  the  center  of  the  station 
to  be  observed  upon,  and  a  ray  of  sunlight  is  thrown  through  a  small  hole 
in  a  board  nailed  to  a  stake  10  or  15  feet  away,  and  in  the  direction  of  the 
observer  at  the  distant  station.     This  form  has  the  advantage  of  simplicity. 


HELIOTEOPES. 


53 


as  the  simplest  backwoodsman  can  manage  it;  a,nd  the  triangulator  can 

firmly  fix  all  range  stakes  upon  one  visit  to  the  station,  and  be  sure  of  seeing 

the  flash  as  he  observes  from  each  of  the  surrounding  stations  in  turn. 

Two  other  forms  are  m  use,  the  Coast   Survey  type  and  the  Steinheil. 

See  Figs.  5  and  6.     The  former  consists  of  a  telescope  which  is  provided 

with  a  screw  for   fastening  it   into    any  con- 
venient support  or  upon  the  theodolite.     Upon 

the  telescope  is  a  mirror  and  two  rings,  the  axis 

of  the  rings  as  well  as  the  center  of  support  of 

the  mirror  being  parallel  to  the  line  of  sight 

of  the  telescope.     The  telescope  being  directed 

upon  the  observing  station,  the  mirror  is  so 

turned  as  to  reflect  the  sunlight  through  the 

rings  and  necessarily  to  the  observing  station. 

In  many  cases  the  use  of  a  second  mirror  is 

necessary,  owing  to  the  relative  position  of  the 

two  stations  and  the  sun,  and  such  a  mirror 

forms  a  part  of  the  outfit.    This  form  is  little 

used,  on  account  of  its  liability  to  get  out  of 

adjustment.     The  Steinheil  heliotrope  is  ac  om- 

pact  little  instrument,  which  can  be  carried  in  a 

case  like  a  pair  of  field  glasses.     It  consists  of 

a  small  sextant  mirror,  the  two  surfaces  of  which 

are  as  nearly  absolutely  parallel  as  possible. 

This  mirror  has  a  small  hole  in  the  center  of 

the  reflecting  surface.     Below  this  central  hole 

is  a  small  lens  in  the  shaft  carrying  the  mirror,  and  below  the  lens  is 

some  white  reflecting  material,  as  plaster  of  Paris.    The  mirror  is  so  mounted 

that  it  has  four  different  motions,  two  about  its  horizontal  axis  and  two 

about  its  vertical  axis,  each  of  which  can  be  separately  bound  or  controlled 

by  clamps  or  friction  movements.     To  use  the  Steinheil,  it  is  screwed  into 

some  wooden  upright,  as  the  side  of  a  tree,  in  suc\i  a  position  that  the  main 

axis  carrying  the  lens  and  plaster  of  Paris  reflector  shall  be  parallel  to  the 

sun's  rays.    The  observer  standing  behind  the  mii'ror  receives  from  the  rear 


Fig.  6.— Steinheil  Heliotrope. 


54  A  MANUAL  OF  TOPOGKAPHIO  METHODS. 

surface  of  the  glass  a  reflection  of  the  sun,  producing  an  imaginary  sun. 
The  mirror  should  not  be  moved  until  this  imaginary  sun,  moving  with  it, 
appears  to  rest  on  the  object  to  which  the  flash  is  to  be  cast,  as  the  hill  on 
which  the  triangulator  is  standing.  As  both  surfaces  of  the  mirror  are  par- 
allel, the  true  reflected  rays  of  the  sun  from  the  surface  of  the  mirror  will 
also  be  cast  on  the  object  sighted  to. 

This  instrument  is  in  great  favor,  especially  with  the  Western  parties, 
where  portability  is  a  matter  of  moment,  first,  because  it  is  light  and  con- 
venient to  carry  and  use,  and  second,  because  there  are  no  movable  parts 
to  get  out  of  adjustment  by  jarring*.  This  latter  is  a  serious  defect  in  the 
Coast  Sm-vey  instrument,  since  unless  frequently  tested  the  two  rings  may 
have  moved,  thus  causing  the  reflection  to  be  cast  out  of  parallelism  with 
the  line  of  sight  of  the  telescope. 

The  use  of  heliotropes  presupposes  the  employment  of  men  to  operate 
them,  thus  increasing  materially  the  expense  of  the  work.  Misunderstand- 
ings continually  arise  between  the  heliotropers  and  the  observer,  causing 
vexatious  delays,  and  therefore  their  employment  should  be  avoided  when- 
ever possible. 

THEODOLITES  FOR  TRIANGULATION. 

Several  instruments  differing  widely  in  power  and  degree  of  accuracy 
have  been  in  use  for  the  measurement  of  angles  in  the  primary  triangula- 
tion.  Formerly  theodolites  having  circles  6,  7,  8,  10,  and  11  inches  in 
diameter  and  reading  by  vernier  to  10  seconds  were  employed,  and  the 
results  were  reduced  and  adjusted  by  Least  Squares.  Subsequently,  it 
appeared  desirable  to  employ  a  higher  class  of  instruments  and  thus  obtain 
more  accurate  results,  which  would  render  unnecessary  this  tedious  adjust- 
ment. Pursuant  to  this  decision  the  use  of  these  vernier  theodolites  has 
been,  in  the  main,  discontinued,  and  theodolites  having  8-inch  circles,  read- 
ing by  micrometer  microscopes,  have  been  substituted  almost  universally 
in  the  primary  work. 

One  of  these  theodolites  is  represented  in  PI.  v  and  Fig.  7. 

The  circle,  as  was  above  stated,  has  a  diameter  of  8  inches,  and  is  sub- 
divided to  10  minutes.     The  object  glass  is  2  inches  in  diameter  and  its 


,   GEOLOGICAL  SURVEY 


EIGHT-INCH  THEODOLITE  AND  TRIPOD. 


THEODOLITE.  55 

focal  distance  is  16^  inches.     The  telescope  with  the  eyepiece  commonly 
used  has  a  -power  of  about  30  diameters. 

The  circle  is  read  by  means  of  two  microscopes,  placed  opposite  one 
another.  Within  the  field  of  the  microscope  is  a  comb  stretching  over  the 
space  of  20  minutes.  This  comb  has  ten  teeth,  divided  into  two  parts  by 
a  depression,  each  corresponding  to  2  minutes.  Parts  of  a  minute  down  to 
2  seconds  are  read  by  means  of  a  micrometer  screw  moving  a  pair  of  fine ' 
tkreads  in  the  field  of  the  microscope. 


/ 


Fig.  7.— Eight-inch  Theodolite,  detail. 
INSTRUCTIONS  FOR  THE  MEASUREMENT  OF  HORIZONTAL  ANGLES. 

The  following  general  precautions  should  be  observed  in  the  measure- 
ment of  all  horizontal  angles  in  the  primary  triangulation.     . 

The  instrument  should  have  a  stable  support,  which  may  be  a  stone 
pier,  a  wooden  post,  or  a  good  tripod.  If  a  portable  tripod  is  used,  its  legs 
should  be  set  firmly  in  the  ground. 

The  instrument  should  be  protected  from  the  direct  rays  of  the  sun  by 
means  of  an  umbrella,  or  a  piece  of  canvas  like  a  tent  fly.  It  should  also 
be  shielded  from  winds  which  ma}-  jar  or  twist  either  it  or  its  support. 

The  foot  screws  of  the  instrument  after   it  is  leveled  for  work  should 


56  A  MANUAL  OF  TOPOGKAPHIC  METHODS. 

be  tight]}-  clamped.  Looseness  of  the  foot  screws  and  tripod,  is  a  common 
source  of  error,  especially  witli  small  instruments. 

The  alidade,  or  part  of  the  instrument  carrying  the  telescope  and 
verniers  or  microscopes,  should  move  freely  on  the  vertical  axis.  Clamps 
should  likewise  move  freely  when  loosened.  Whenever  either  of  these 
moves  tightly,  the  instrument  needs  cleaning,  oiling,  or  adjusting. 

The  observer  should  always  have  a  definite  preliminary  knowledge  of 
the  objects  or  signals  observed.  The  lack  of  it  may  lead  to  serious  error 
and  entail  cost  nnich  in  excess  of  that  involved  in  getting  such  knowledge. 

Great  care  should  be  taken  to  insure  correctness  in  the  degrees  and 
minutes  of  an  observed  angle.  The  removal  of  an  ambiguity  in  them  is 
sometimes  a  troublesome  or  expensive  task. 

The  errors  to  which  measured  angles  are  subject  may  be  divided  into 
two  classes — viz.,  first,  those  dependent  on  the  instrument  used,  or  instru- 
mental errors;  and  second,  those  arising  from  all  other  sources,  Avhich,  for 
the  sake  of  distinction,  may  be  called  extra-instrumental  errors. 

The  best  instrimients  are  more  or  less  defective,  and  all  adjustments 
on  which  precision  depends  are  liable  to  derangement.  Hence  the  general 
practice  of  arranging  observations  in  such  a  manner  that  the  errors  due  to 
instrumental  defects  will  be  eliminated  in  the  end  results.  The  principal 
errors  of  this  kind  and  the  methods  of  avoiding  their  effects  are  enumerated 
below. 

Measurements  made  with  a  graduated  circle  are  subject  to  certain  sys- 
tematic errors  commonly  called  periodic.  Certain  of  these  errors  are  always 
eliminated  in  the  mean  (or  sum)  of  the  readings  of  the  equidistant  verniers 
or  microscopes,  and  both  of  the  latter  should  be  read  with  equal  care  in 
precise  work.  Certain  other  errors  of  this  class  are  not  eliminated  in  the 
mean  of  the  microscope  readings,  and  these  only  need  consideration.  Their 
effect  on  the  mean  of  all  the  measures  of  an  angle  may  be  rendered  insig- 
nificant by  making  the  number  of  individual  measures  with  the  circles  in 

each  of  n  equidistant  positions  separated  by  an  interval  equal  to  ^ —  where 

m  is  the  number  of  equidistant  verniers  or  microscopes.     Thus,  if  w?=:2, 

1  80° 
the  circle  should  be  shifted  after  each  measure  by  an  amount  equal  to 


INSTRUCTIONS.  57 

which,  for  example,  is  45°  for  «  — 4  aud  30°  if  n=Q.  The  degree  of  ap- 
proximation of  this  elimiuation  increases  rapidly  with  n.  (For  specifications 
as  to  particular  instruments  see  "Number  of  sets  required  and  astronomical 
azimuths"  below.)  The  effect  of  this  class  of  errors  is  always  nil  on  an 
angle  equal  to  the  angular  distance  between  consecutive  microscopes  or  a 
multiple  thereof  Other  things  equal,  therefore,  we  would  expect  the  measures 
of  such  special  angles  to  show  less  range  than  the  measures  of  other  angles. 

Besides  the  instrumental  errors  of  the  periodic  class,  there  are  also 
accidental  errors  of  graduation.  These  are  in  general  small,  however,  in 
the  best  modern  circles  and  their  effect  is  sufficiently  eliminated  by  shifting 
the  circle  in  the  manner  explained  under  "Periodic  errors"  above. 

The  effect  of  an  error  of  collimation  on  the  circle  reading  for  any 
direction  varies  as  the  secant  of  the  altitude  of  the  object  observed.  The 
effect  on  an  angle  between  two  objects  varies  as  the  difference  between  the 
secants  of  their  altitudes.  This  effect  is  eliminated  either  by  reversing  the 
telescope  in  its  Ys,  or  by  transmitting  it  without  changing  the  pivots  in  the 
Ys,  the  same  number  of  measures  being  obtained  in  each  of  the  two  posi- 
tions of  the  telescope.  The  latter  method  is  the  better  one,  especially  in 
determining  azimuth,  since  it  eliminates  at  the  same  time  errors  due  to 
inequality  of  pivots  and  inequality  in  height  of  the  Ys. 

The  effect  of  the  error  of  inclination  on  the  circle  reading  for  any 
direction  varies  as  the  tangent  of  the  altitude  of  the  object  observed.     If 
the  inclination  is  small,  as  it  may  always  be  by  proper  adjustment,  its  effect 
will  be  negligible  in  most  cases.     But  if  the  objects  differ  much  in  altitude, 
as  in  azimuth  work,  the  inclination  of  the  axis  must  be  carefully  measured 
with  the  striding  level,  so  that  the  proper  correction  can  be  applied.     The 
following  formula  includes  the  corrections  to  the  circle  reading  on  any 
object  for  collimation  and  inclination  of  telescope  axis: 
c  sec  /<  +  b  tan  h; 
c  zz  collimation  in  seconds  of  arc, 
b  zz  inclination  of  axis  in  seconds  of  arc, 
h  zr  altitude  of  object  observed. 

Parallax  of  wires  occurs  when  they  are  not  in  the  common  focal  plane 
of  the  eyepiece  and  objective.  It  is  detected  by  moving  the  eye  to  and 
fro  sidewise  while  looking  at  the  wires  and  image  of  the  object  observed. 


58  A  MANUAL  OP  TOPOGRAPHIC  METHODS. 

If  the  wires  appear  to  move  in  the  least,  ah  adjustment  is  necessary.  The 
eyepiece  should  always  be  first  adjusted  to  give  distinct  vision  of  the  cross 
wires.  This  adjustment  is  entirely  independent  of  all  others  and  requires 
only  that  light  enough  to  illuminate  the  wires  enter  the  telescope  or  micro- 
scope tube.  This  adjustment  is  dependent  on  the  eye  and  is  in  general 
different  for  different  persons.  Hence  maladjustment  of  the  eyepiece  can 
not  be  corrected  by  moving  the  cross  wires  with  reference  to  the  objective. 
Ha^ang  adjusted  the  eyepiece,  the  image  of  the  object  observed  may  be 
brought  into  the  plane  of  the  cross  wires  by  means  of  the  rack-arid-pinion 
moveuient  of  the  telescope.     A  few  trials  will  make  the  parallax  disappear. 

When  circles  are  read  by  micrometer  microscopes  it  is  customary  to 
have  them. so  adjusted  that  an  even  number  of  revolutions  of  the  screw  will 
carry  the  wires  over  the  image  of  a  graduation  space.  If  the  adjustment 
is  not  perfect,  an  error  of  run  will  be  introduced.  This  may  in  all  cases  be 
made  small  or  negligible,  since  by  means  of  the  independent  movements  of 
the  whole  microscope  and  the  objective  with  respect  to  the  circle,  the  image 
may  be  given  any  required  size.  In  making  this  adjustment  some  standard 
space,  or  space  whose  error  is  known,  should  be  used.  At  least  once  at  each 
station  where  angles  are  read,  observations  should  be  made  for  run  of 
micrometers.  Au  example  of  such  readings  is  given  under  sample  of  field 
notes  below. 

Tangent  and  micrometer  screws  should  move  freely,  but  never  loosely. 
In  making  a  jjointing  with  the  telescope  the  tangent  screw  should  always 
move  against  or  push  the  opposing  spring.  Likewise,  bisections  with  the 
rhicrometer  wires  should  be  made  always  by  making  the  screw  pull  the 
micrometer  frame  against  the  opposing  spring  or  springs. 

Extra  instrumental  errors  may  be  divided  into  four  classes — namely, 
errors  of  observation,  errors  from  twist  of  tripod  or  other  support,  errors 
from  centering,  and  errors  from  unsteadiness  of  the  atmosphere. 

Barring  blunders  or  mistakes,  the  errors  of  observation  are  in  general 
relatively  small  or  unimportant.  With  practiced  observers  in  angular  meas- 
urements, such  errors  are  the  least  formidable  of  all  the  unavoidable  errors, 
and  then'  elimination  in  the  end  results  is  usually  well  nigh  perfect.  The 
recognition  of  this  fact  is  very  important,  for  observers  are  prone  to  attribute 


INSTEUCTIONS.  59 

unexpected  discrepancies  to  bad  observation  rather  than  to  their  much  more 
probable  cause.  After  learning-  how  to  make  good  observations  the  observer 
should  place  the  utmost  confidence  in  them,  and  never  yield  to  the  tempta- 
tion of  changing  them  because  they  disagree  with  some  preceding  observa- 
tions. Such  discrepancies  are  in  general  an  indication  of  good,  rather  than 
poor,  work. 

Stations  or  tripods  which  have  been  unequally  heated  by  the  sun  or 
other  source  of  heat  usually  twist  more  or  less  in  azimtith.  The  rate  of 
this  twist  is  often  as  great  as  a  second  of  arc  per  minute  of  time,  and  it  is 
generally  nearly  uniform  for  intervals  of  ten  to  twenty  minutes.  The  effect 
of  twist  is  to  make  measured  angles  too  great  or  too  small  according  as  they 
ai-e  observed  by  turning  the  microscopes  in  the  direction  of  increasing  gradua- 
tion or  in  the  opposite  direction.  This  effect  is  well  eliminated,  in  g-eneral, 
in  the  mean  of  two  measiu-es,  one  made  by  turning  the  microscopes  in  the 
direction  of  increasing  graduation  and  followed  immediately  by  turning  the 
microscopes  in  the  opposite  direction.  Such  means  are  called  combined 
measures  or  combined  results,  and  all  results  used  should  be  of  this  kind. 
As  the  uniformity  in  rate  of  twist  can  not  be  depended  on  for  any  considera- 
ble interval,  the  more  rapidly  the  observations  on  an  ang'le  can  be  made 
the  better  will  be  the  elimination  of  the  twist.  The  observer  should  not 
wait  more  than  two  or  three  minutes  after  pointing  on  one  signal  before  point- 
ing on  the  next.  If  for  any  reason  it  should  be  necessary  to  wait  longer,  it 
will  be  best  to  make  a  new  reading  on  the  first  signal. 

The  precision  of  centering  an  instrument  or  signal  over  the  reference  or 
geodetic  point  increases  in  importance  inversely  as  the  length  of  the  ti'iangu- 
lation  lines.  Thus,  if  it  is  desired  to  exclude  errors  from  this  source  as  small 
as  a  second,  one  must  know  the  position  of  the  instrument  within  one-third 
of  an  inch  for  lines  a  mile  long',  or  within  6  inches  for  lines  20  miles  long-. 
The  following  easily  remembered  relations  will  serve  as  a  guide  to  the  re- 
quired precision  in  any  case : 

1  second  is  equivalent  to  0.3  inch  at  the  distance  of  1  mile. 

1  second  is  equivalent  to  3.0  inches  at  the  distance  of  10  miles. 

1  second  is  equivalent  to  6.0  inches  at  the  distance  of  20  miles. 

1  minute  is  equivalent  to  1.5  feet  at  the  distance  of  1  mile. 


60  A  MANUAL  OF  TOrOaKAPHlO  METHODS. 

The  notes  should  always  state  explicitlj-  where  the  mstrument  aud 
signals  are  and  give  their  coordinates  (preferably  polar  coordinates)  if  they 
ai'e  not  centered. 

Objects  seen  tlu-ough  the  atmosphere  appear  almost  always  unsteady, 
and  sometiuies  this  unsteadiness  is  so  great  as  to  render  the  identity  of  the 
object  doubtful.  The  unsteadiness  is  usually  greatest  during  the  middle  of 
the  day.  It  generally  subsides  or  ceases  for  a  considerable  period  between 
2  p.  m.  and  sundown.  There  is  also  frequently  a  short  interval  of  quietude 
about  sunrise,  and  on  cloudy  days  many  consecutive  hours  of  steady 
atmosphere  may  occur.  For  the  best  woi-k,  observations  should  he  made 
only  when  the  air  causes  small  or  imperceptible  displacements  of  signals. 
In  applying  this  rule,  however,  the  observer  must  use  his  discretion.  Errors 
of  pointing  increase  rapidly  with  increase  of  unsteadiness,  but  it  will  fre- 
quently happen  that  time  may  be  saved  by  counterbalancing  errors  from  this 
source  by  making  a  greater  number  of  observations.  Thus,  if  signals  are 
fairly  steady  it  may  be  economical  to  make  double  the  number  of  observa- 
tions rather  than  wait  for  better  conditions. 

The  best  results  in  a  triangulatiou  are  to  be  obtained  by  measuring  the 
angles  separately  and  independently.  Thus,  if  the  signals  in  sight  around 
the  horizon  are  in  order  A,  B,  C,  etc.,  the  angles  A  to  B,  B  to  C,  etc.,  are  by 
this  method  observed  separately;  and  whenever  there  is  sufficient  time  at 
the  observer's  disposal  this  method  should  be  followed. 

Besides  measuring  single  angles,  it  is  desirable  to  measure  independ- 
ently combined  angles — i.  e.,  angles  which  consist  of  the  sum  of  two  or  more 
single  angles.  Thus,  supposing  O  to  be  the  observing  station  and  A,  B,  and 
C  stations  sighted  on,  the  observer  sliould  measure  not  only  the  angles  AOB 
and  BOC,  but  the  combined  angle  AOC.  This  is  necessary  not  only  because 
this  angle  may  be  used  directly  in  the  triangulation,  but  it  will  be  needed  in 
fonning  conditions  for  adjusting  tlie  angles  about  the  observing  station,  or 
the  station  adjustment,  as  it  is  called. 

In  order  to  secure  the  elimination  of  the  errors  mentioned  above,  the 
following  programme  must  be  strictly  adhered  to: 

Pointing  on  A  and  readings  of  both  microscopes. 

Pointing  on  B  and  readings  of  both  microscopes. 


INSTRUCTIONS.  61 

Transit  telescope  and  tnrn  microscopes  180°. 

Pointing  on  B  and  readings  of  both  microscopes. 

Pointing  on  A  and  readings  of  both  microscopes. 

1 80° 
Shift  circle  by and  proceed  as  before  until  n  such  sets  of  measures 

have  been  obtained. 

Then  measure  the  angles  B  to  C,  C  to  D,  etc.,  including  the  angle 
necessary  to  close  the  horizon,  in  the  same  manner. 

A  form  for  record  and  computation  of  the  results  is  given  below. 

When  repeating  instruments  are  used,  the  same  programme  will  be  fol- 
lowed except  that  there  should  be  five  pointings  instead  of  one  on  each  of 
A  and  B,  the  circle  being  read  for  the  first  pointing  on  A  and  the  fifth  on 
B,  and  again  for  the  sixth  pointing  on  B  and  the  tenth  on  A. 

The  impoi-tance  of  having  the  measin-es  of  a  set  follow  in  quick  succes- 
sion must  be  constantly  borne  in  mind.  Under  ordinarily  favorable  condi- 
tions an  observer  can  make  a  pointing  and  read  the  microscopes  once  a 
minute,  and  a  set  of  five  reijetitions  should  be  made  in  five  minutes  or  less. 

When  several  stations  or  signals  are  visible  and  a  nonrepeating  instru- 
ment is  used,  time  may  be  saved  without  material  loss  of  precision  in  the 
angles,  by  observing  on  all  the  signals  successively  according  to  the  follow- 
ing programme,  the  signals  being  supposed  in  the  order  A,  B,  C,  etc.,  as  above. 

Pointing  on  A  with  microscope  readings. 

Pointing  on  B  with  microscope  readings. 

Pointing  on  C  with  microscope  readings. 

Pointing  on  A  with  microscope  readings. 
Transit  telescope  and  turn  microscopes  180°. 
Pointing  on  A  with  microscope  readings. 
Pointing  on  B  with  microscope  readings. 
Pointing  on  C  with  microscope  readings. 

Pointing  on  A  with  microscope  readings. 

180 
Shift  circle  by and  proceed  as  before  until  n  such  sets  have  been 

obtained. 


02  A  MANUAL  OF  TOPOGRArHIC  METHODS. 

The  angles  A  to  B,  B  to  C,  etc.,  read  in  tlii.s  way  may  be  computed  as 
iu  the  first  method,  always  combining  the  measure  A  to  B  with  the  immedi- 
ately succeeding  measure  B  to  A  to  eliminate  twist.  There  is  a  theoretical 
objection  to  this  process  of  deriving  angles  founded  on  the  fact  that  they 
are  not  independent,  but  in  secondary  work  this  objection  may  be  ignored 
as  of  little  weight. 

For  the  11 -inch  theodolite  and  for  the  .new  8-inch  instruments  made 
by  Fauth  &  Co.,  all  of  which  read  by  micrometer  microscopes,  four  (4)  sets 
of  measures  on  as  many  different  parts  of  the  circle  will  be  required ;  and 
for  the  repeating  theodolite  six  (6)  sets  of  measures  will  be  requued,  all 
measures  being  made  according  to  the  programmes  given  above. 

Under  ordinary  circumstances  and  with  due  care  in  centering,  angles 
measured  as  specified  above  should  show  an  average  error  of  closure  of  the 
triangles  not  exceeding  5". 

Under  specially  unfavorable  conditions  the  number  of  sets  of  measures 
should  be  increased,  care  being  always  taken  to  shift  the  circle  so  as  to 
eliminate  periodic  eiTors. 

The  practice  of  starting  the  measurement  of  an  angle  or  series  of 
angles  with  the  microscopes  reading  0°  and  180°,  90°,  and  270°,  etc.,  must 
be  avoided;  otherwise  the  errors  of  these  particular  divisions  will  affect 
many  angles.     In  shifting  the  circle  it  is  neither  necessarj^  nor  desirable  to 

1  on 

have  the  new  positions  differ  from  the  preceding  one  by  exactly .     A 

difference  of  half  a  degree  either  way  is  unimportant  as  respects  periodic 
errors,  and  it  is  advantageous  to  have  the  minutes  and  seconds  differ  for  the 
different  settings. 

Field  notes  should  be  clear  and  full.  The  date,  place,  name  and  num- 
ber of  instrument  used,  and  the  names  of  observer  and  recorder  should  be 
recorded  at  the  beginning  of  each  day's  work  at  a  station.  The  positions 
of  the  instrument  and  signals  observed  should  be  defined  either  by  a  full 
statement  or  reference  to  such  in  each  day's  notes.  The  time  of  observa- 
tions should  be  noted  at  intervals  to  show  that  the  instrument  does  not 
stand  too  long  between  pointings. 


ORGANIZATION^  OF  PARTIES. 


63 


When  mistakes  are  made  in  the  record,  the  defective  figures  should  not 
be  erased,  but  simply  crossed  out,  and  an  explanation  furnished  in  the  col- 
umn of  remarks.  Grreat  care  should  be  taken  not  only  to  avoid  "cooking" 
or  "doctoring"  notes,  but  to  avoid  suspicion  thereof. 

The  following  example  of  form  of  record  is  taken  from  the  primary 
triangulation  executed  in  1889  in  western  Kansas: 

Record  of  measurement  of  horizontal  angle. 

tliviaiou  of  micrometer 


Station. 

Micr.  A.               Micr.  B, 

Mea 

n  reading. 

Angle. 

Mean. 

Telescope  direct. 

°       '      Div.         °       '     Div. 

93     12     11.3        273     12    09.9 
129    41     11.9        309    41     13.2 
129    41    16.6       309    41    12.1 

93    12    10.6       273    12    09.1 

Telescope  reversed. 
138    27     03.2        318     28     28.0 
174    66     03.8        354     65    28.9 
174    56     06.  2       354     55    29. 5 
138    27     05.^2        318     26     27.4 

Telescope  reversed. 
1S3    07     03.0           3     06    27.2 
219    36     05.0         39     35     29.8 
219    36     08.  1         39    35     29. 5 
183    07     06.4           3     06     28.1 

93 
129 
129 

93 

138 

174 
174 
138 

183 
219 
219 
183 

228 
264 
264 
228 

1'2    21.2 
41     25.  1 
41     27.7 
12    19.7 

27    01.2 
50    01.7 
56    06.7 
■27    02.6 

07    00.2 
36    04.8 
36    07.6 
07    04.5 

24    50.7 
63    63.5 
63    .57.2 
24    61.4 

36    29     03.9 
08.0 

00.5 
03.1 

04.6 
03.1 

02.8 
05.8 

05.9 

01.8 
03.9 
04.3 

Telescop 
228    21     28. 1 

e  direct. 
.48    24    22.6 

264    53     27.4 
264    64    01.1 
228    24    29.3 

84     53     26. 1 
84     53     26.1 
48     24     22.1 

Newt 

41  15" 
=360  29'  03". ! 


^  Instrument  over  cetter  of  station. 


ORGANIZATION  OF  PARTIES  AND  PROSECUTION  OF  WORK. 

A  party  for  carrying  on  primary  triangulation  usually  comprises  only 
the  chief  and  an  assistant,  with  the  addition  of  a  driver  and  cook,  in  case  the 
party  is  living  in  camp.  Frequently,  however,  a  man  is  employed  to  super- 
intend the  construction  of  signals,  and  it  is  generally  found  economical  to  em- 
ploy such  a  man.  The  chief  of  the  party  is  expected  to  select  the  stations 
and  direct  the  forms  of  signals  to  be  erected,  and  to  measure  angles.  In  a 
mountainous  country  the  selection  of  stations  is  usually  a  simple  matter. 
From  the  summit  of  a  mountain  the  chief  of  a  party  may  be  able  to  select 
stations  for  considerable  distances  ahead  and  to  order  the  erection  of  signals, 
turning  over  to  the  man  employed  for  that  purpose  the  business  of  erecting 


04  A  MANUAL  OF  TOPOGEAPHIO  METHODS. 

tlieni.  On  the  other  haucl,  in  a  densely  wooded  region  such  as  the  Cumber- 
hmd  phitean,  where  the  summits  have  approximately  the  same  elevation,  the 
selection  of  stations  is  an  extremely  difficult  matter,  requiring-  great  ability 
and  experience  and  involving  an  immense  amount  of  labor.  In  such  a  region 
the  chief  of  party  finds  it  necessary  to  travel  great  distances,  visit  many  hills, 
and  even  has  to  climb  to  the  summits  of  the  highest  trees,  in  order  to  select 
iutervisible  stations. 

The  selection  of  stations  must  be  kept  in  advance  of  the  reading  of 
angles,  but  it  is  not  advisable  to  keep  it  too  far  ahead,  on  account  of  the 
danger  of  the  destruction  of  signals  before  angles  have  been  read  upon  them. 
Therefore,  the  chief  of  a  part}^  finds  it  necessary  to  alternate  between  the 
two  kinds  of  work,  selecting  and  preparing  three  or  four  stations,  then  re- 
turning and  measuring  the  angles. 

"When  it  is  necessary  to  use  heliotropes,  the  party  has  necessarily  to  be 
increased  by  one  man  for  each  heliotrope  employed.  The  proper  manage- 
ment of  such  a  party  then  becomes  a  matter  calling  for  the  exercise  of  much 
judgment  on  the  part  of  the  triangulator.  If  it  is  convenient  for  the  chief  of 
party  to  place  each  heliotroper  before  observing  angles,  and  to  show  them 
where  to  direct  their  instruments,  men  of  ordinary  intelligence  may  be  em- 
ployed and  the  work  is  one  calling  rather  for  time  than  skill.  Where,  how- 
ever, the  party  is  moving  almost  daily,  the  observer  and  heliotropers  occu- 
pjnng  a  different  station  nearly  every  day,  as  is  possible  in  the  dry  and  clear 
atmosphere  usually  prevailing  in  the  West,  the  chief  of  party  has  to  arrange 
a  schedule  for  each  man,  showing  the  order  in  which  he  is  to  occupy  the 
stations  and  in  what  direction  he  is  to  flash  from  each.  The  heliotroper 
must  be  a  man  having  some  topographic  and  technical  skill,  so  that  he  may 
find  his  point,  set  up  on  center  and  direct  his  flashes  to  the  right  place, 
besides  exercising  a  goodly  amount  of  common  sense  judgment.  A  simple 
code  of  signals  being  agreed  upon,  it  then  becomes  an  easy  matter  for  the 
triangulator  to  let  the  heliotropers  know  that  the  work  is  completed,  when 
they  at  once  move  to  the  next  designated  station. 


REDUCTION  OP  TRIANGULATION. 


65 


REDUCTION   OF  PRIMARY  TRIANGULATION. 
*KBDU(!TION    TO    CBNTEE. 

In  case  any  station  was  occupied  off  center,  the  directions,  as  read  must 
first  be  reduced  to  center.     In  the  diagram,  let  x  be  the 


point  occupied;  y,  the  station,  r  the  distance  between  them,  A  the  point  to 
which  the  direction  is  laid  and  the  angle  at  that  point,  and  R  its  distance, 
approximately  known.  Then,  from  the  relations  between  the  sides  and  the 
angles  of  the  triangle, 

R  :  r  : :  sin  x  :  sin  A 

r  sin  X 


A-- 


-  and  A  zz  (in  seconds) 


R     ""^"  — V ^^"^Rsinl" 

correction  in  seconds  of  arc. 
The  following  example  taken  from  the  triangulation  in  Kansas  will 
serve  to  illustrate  the  form  of  effecting  this  reduction.     The  references  are 
to  the  diagram  on  page  67. 

Reduction  to  center  of  station  at  Walton  A- 
[See  explanation:  Appendix  No.  9,  page  167,  U.  S.  Coast  and  Geodetic  Survey  report  for  1882.] 

distance,  inst.  to  center^ '.48  log  =    9.6812 
log  feet  to  meters  =    0.  5160 

distance,  inst.  to  center  log  meters  =    9. 1652  =  log  r. 


Direction. 

xton 

7°. 

xtoo 
73°. 

X  to  p 

105°. 

X  to  q 

185°. 

X  tor 
273°. 

X  to  s 
306°. 

9.  0859 
6.  9.321 
9. 1652 
5.3144 

9.  9806 
5.  9182 
9. 1652 
5.  3144 

9.  9849 
6. 4228 
9. 1652 
5.3144 

8. 9403 
6. 2434 
9. 1652 
5.3144 

9. 9994 
6,  0070 
9. 1652 
5.  3144 

9.  9080 
6.  2514 
9.1652 
5.3144 

Correction  to  direction 

9.  4976 
0",31 

0.  3784 
2".  39 

0.  8873 
7".  71 

9. 6633 
0".  46 

0.48(i9 
3".  06 

0.  6390 
4".  36 

Correction  to  ang 


,  a  =  Jl,  to  0  —0.  31  +2.  39  =  +2.  08 
6  =  o  to  p  — 2,  39  +7.  71  =  +5.  32 
a  —  n  to  p  —0.  31  +7.  71  =  +7.40 
c—p  taq  —7.  71  —0.  46  =  —8. 17 
d!  =  o  to  r  +0.  46  —3.  06  =  —2.  60 
e  =  r  to  s  +3.  06  —4.  36  =  —1.  30 
ft  =  n  to  s  +0.  46  —4.  36  =  -3.  90 
/  =  s  to  n  +4.  36  +0.  31  =  +4.  67 


The  angles  are  measured  on  a  spherical  surface  and  the  sum  of  the 
three  measured  angles  of  each  triangle  should  equal  180°  plus  the  spher- 


MON    XXII- 


66 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


ical  excess.  The  latter  need  be  computed  aud  subtracted  from  the  sum.  of 
the  angles,  however,  ouly  for  the  purpose  of  testing  the  accuracy  of  closure 
of  the  triangle,  as  in  the  reduction  the  angles  are  treated  as  plane  angles. 
When  the  area  of  the  triangle  is  large,  the  spherical  excess  in  seconds  (E) 
should  be  computed  by  the  equation: 


E  — 


S 
r^  sin  1 


where  S  =z  the  area  of  the  triangle  in  square  miles,  and  r  the  radius  of 
curvature  of  the  earth  in  miles.  When  the  triangle  (being  within  the 
United  States)  has  an  area  less  than  500  square  miles,  r  may  be  assumed 
as  constant,  and  the  spherical  excess  may  be  obtained  by  dividing  the  area 
in  square  miles  by  75.5. 

The  next  step  is  the  adjustment  of  the  angles  about  the  observing  sta- 
tion, or  the  station  adjustment,  as  it  is  called.  Referring  to  the  diagram, 
which  represents  the  angles  read  at  Walton  station,  in  Kansas,  it  is  seen  that 
eight  angles  were  measured  as  follows — 


Obs.  angle. 

Station 
adjust- 
ment. 

Correc- 
tion to 
center. 

Angles  locally 

adjusted  and 

reduced  to 

center. 

65 
31 

45 
47 

28.37 
58.50 

+.51 
+.52 

+2.08 
+5.32 

65    45    30.96 
31    48    04.34 

Sum ^= 

97 
97 

33 
38 

26.87 
28.39 

97    33     35.30 
97    33     35.30 

—.49 

+7.40 

Bifference =^ 

—1.52 

00.00 

—.56 
—.56 

—2.60 
—1.30 

87 
34 

44 
00 

57.41 
03.35 

87    44    54.25 
34    00    01.49 

Sum = 

121 
121 

44 
44 

60.76 
59.05 

121    44    55.74 
121     44     55.74 

+.59 

—3.90 

+1.71 

00.00 

+.02 
—.49 
+  .02 
+  .59 

44.67 
+7.40 
—8.17 
-3.90 

61 

97 
79 
121 

09 
33 
32 
44 

26,17 
28.39 
06.25 
59.05 

61     09    30.86 
97    33    35.30 
79    31    58.10 
121    44    55.74 

Sum =^ 

359 

59 

59.86 
—0.14 

360     00    00.00 
00.00 

Of  these  a-\-b  should  =:  g,  d-\-e  should  =  h,  and  g -\-  c  -\- h -\- f  should 
=  360°.  Thus  are  formed  in  this  case  three  conditions  affecting  eight 
unknown  quantities.     The  method  by  which  are  found  the  corrections  which 


EBDUCTION  OF  TEIANGULATION. 


67 


fulfill  these  conditions  is  that  known  as  the  method  of  Least  Squares.     It  is 
umiecessaiy  to  explain  the  theory  of  this  method,  but  only  to  show  how  it 


is  applied  in  the  class  of  cases  under  consideration,  which  can  best  be 
done  by  tracing  a  case  through.  There  are  here  three  equations  of  condi- 
tions, as  follows: 

(1)  a-\-h—[/—l".52         =0 

(2)  f?+e-/i+l".71  rrO 

(3)  ,/■  +  //  +  c  +  /i  -  0".14  =  0 

in  which  the  letters  represent,  not,  as  in  the  diagram,  angles,  but  unknown 
con-ections  to  the  angles.  The  coeflicient  of  each  of  these  corrections  is 
unity.  Arrange  them  in  tabular  form,  the  numbers  at  the  top  referring  to 
the  equations,  thus  forming  what  is  called  a  table  of  correlates.  Now  mul- 
tiply each  coefficient  by  itself  and  every  other  in  the  same  horizontal  line, 
and  sum  them.     There  result  three  normal  equations,  as  follows : 


a  1 

b  1 

d  1 

e  1 

i'       -'        -^ 


1 

+3 

007/ 

.OOz 

-T 

..'^3  = 

=  0 

2 

+  3.00)/- 

.003 

■fV 

.71  = 

=  0 

a 

—  1 

IMw 

—l.UVy  +i.Mz 

-0' 

.14  = 

=  0 

68 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


These  three  equations  iuvolviug  three  imkiiown  quantities,  are  then 
solved  by  ehmination,  with  results  as  follows: 

(.(;=: +.515 

y  —  —.562 

^  =  +.023 
These  values  can  now  be  substituted  in  the  table  of  correlates,  columns 
1,  2,  3;  the  algebraic  sum  of  hues  a,  h,  c,  cl,  etc.,  giving  corrections  to  the 
angles  a,  b,  c,  d,  etc. 

,  .,  n       Corrections  to 


+.  51S 

+.515 

b 

'+.515 

+  .615 

c 

+.023  • 

+.023 

d 

-.662 
-.562 

—  562 
—.662 

f 

+.023 

+.023 

n 

—.515 

+.033 

—.492 

h 

+.562 

+  .023 

+  .685 

FIGURE  ADJUSTMENT. 

The  measiu-ement  of  the  angles  having  been  executed  by  instruments 
and  methods  much  better  than  the  needs  of  the  map  require,  it  is  not  ordi- 
narily necessary  to  make  any  figure  adjustment,  further  than  an  equal  dis- 
tribution of  the  error  of  each  triangle  among  the  tlu-ee  angles. 

Stillj  as  the  necessity  for  a  more  elaborate  adjustment  may  arise,  a 
description  of  the  method  of  applying  the  least  square  adjustment  to  geo- 
metric figures  in  triangulation  is  here  given,  with  a  simple  example  of  its 
apphcation. 

Each  geometric  figure  in  a  system  of  triangulation  is  composed  of  a 
number  of  triangles.  The  measured  angles  of  each  triangle  should  equal 
180°  plus  the  spherical  excess.  Each  triangle,  therefore,  furnishes  an  equa- 
tion of  condition,  which  is  known  as  an  angle  equation.  The  number  of 
angle  equations  in  any  figure  is  equal  to  the  number  of  closed  triangles 
into  which  it  can  be  resolved.  But  since  certain  of  these  are  a  consequence 
of  the  others,  the  number  of  angle  conditions  which  it  is  desirable  to  intro- 
duce is  less  than  the  number  of  triangles. 

The  number  of  angle  equations  in  any  figure  is  equal  to  the  number 
of  closed  lines  in  the  figure  plus  one,  minus  the  number  of  stations.  Thus, 
in"  a  closed  quadrilateral,  the  number  of  angle  equations  is  6  +  1  —  4  —  3. 


EEDUOTION  OF  TEI ANGULATION.  69 

There  is  another  class  of  conditions,  known  as  side  equations,  which 
can  be  best  explained  by  reference  to  a  figure.  In  the  example,  diagram, 
suppose  the  figure  0,  1,  2,  3  to  represent  the  projection  of  a  pyramid, 
of  which  1,  2,  3  is  the  base  and  0  the  apex.  A  geometric  condition  of  such 
figm-e  is  that  the  sums  of  the  logarithmic  sines  of  the 
angles  about  the  base,  taken  in  one  direction,  must 
equal  the  similar  sums  taken  in  the  other  direction, 
i.  e.,  the  product  of  the  sines  must  be  equal.  In  the 
present  case,  log.  sin  0,  1,  2  +  log.  sin  0,  2,  3  +  log. 
sin  1,  3;  0  should  equal  log.  sin  1,  2,  0  +  log.  sin  2,  3,  0  +  log.  sin  0,  1,  3. 

The  number  of  side  equations  which  can  be  formed  in  any  figure  is 
equal  to  the  number  of  lines  in  the  figure,  plus  3,  minus  twice  the  number 
of  stations  in  it  or  /  +  3  —  2  n.     In  a  quadrilateral,  6  +  3  —  8  r=  1. 

The  numerical  term  in  each  angle  equation  is  the  difi'erence  between 
the  sum  of  the  observed  angles  on  the  one  hand  and  180°  +  the  spherical ' 
excess  on  the  other.     This  is  positive  when  the  sum  of  the  observed  angles 
is  the  greater,  and  vice  versa.     The  coefficients  of  the  unknown  corrections 
are  in  each  case  unity,  unless  weights  are  assigned. 

The  numerical  term  in  each  side  equation  is  the  difference  between 
the  sums  of  the  logarithmic  sines,  taken  in  the  two  directions.  The  coeffi- 
cients of  the  unknown  corrections  are  the  differences  for  one  second,  in  the 
logarithmic  sines  of  the  angles. 

The  method  of  making  up  and  solving  these  equations  and  applying 
the  corrections  to  the  angles  can  best  be  shown  by  means  of  an  example. 
That -here  given  is  the  simplest  case  involving  both  angle  and  side  equa- 
tions, namely,  the  case  of  a  quadi'ilateral.  The  method  of  forming  correla- 
tives and  normal  equations,  and  their  solution,  is  similar  to  that  employed 
in  station  adjustment,  and  therefore  the  details  are  omitted. 

In  the  equations  of  conditions  and  correlatives,  the  angles  are  desig- 
nated by  directions,  to  which  the  corrections  are  finally  applied.  Thus 
the  angle  of  302  is  designated  as  —  3/0  -|-  2/0,  the  sign  —  being  given  to 
the  left-hand  and  the  sign  +  to  the  right-hand  direction. 


70 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

Example  of  figure  adjustment  hy  least  squares,. 


Observed  angles. 

c 

3-0-1 

120 

39     14. 781 

(«)..< 

01-3 

21 

20     17.806 

< 

1-3-0 

37 

54     37.  ISO 

180 

00     09. 767 

ST)herica 

excess 

=  —0.148 

Closure  erroi- 

+  9.619 

( 

0-1-2 

81 

52     51.222 

(h)..l 

1-2-0 

62 

22     38.500 

\ 

2-0-1 

35 

44     45.861 

180 

00     15. 583 
—  0. 189 

Closure  error 

+15.  394 

1-2-3 

91 

28     38.000 

2-3-1 

28 

95     10.360 

3-1-2 

60 

26     33. 410 

180 

00     21. 776 
—  0.  234 

Closure ( 

rror 

+     21.542 

c 

2-3-0 

65 

59    47. 540 

{c)..{ 

3-0-2 

84 

54    28.920 

\ 

0'2-3 

29 

05    59. 600 

180 

00     15. 9C0 
—      0. 193 

Closure  e 

rror 

+     15.767 

Side  equation. 
[Taking  0  as  the  pole.] 


Angle. 

Log.  sines  of 
spherical  angle. 

Tabular 

difference 

for  1". 

Correc- 
tions to 
log.  sines. 

Corrected  log. 

sines  of  spherical 

angles. 

Spherical 
excess. 

Log.  sines  of 

(<2)- 

0.1.2 

0.2.3 

1.3.0 

Sum = 

1.2.0 

2.3.0 

0.1.3 

Sum := 

From  above 

Difference 

9.9956249.7 
9.  6869340.  0 
9.7884705.9 

+3.0 
37.9 
27.0 

11.0 
9.4 
53.7 

-25.0 

—127. 9 

—1.2 

-59.4 
-77.7 
-203.0 

9.  9956224.  7 
9.  6869212. 1 
9.7884704.7 

29.4710141.5 

—.063 
—.065 
—.050 

—.063 
—.064 
-.049 

9.  9956224 
9.  6869210 
9. 7884703 

29.4710295.6 

29. 4710137 

9.  9474437.  5 
9.  9607184. 9 
9.5628859.2 

9.  9474378.  1 
9.  9607107.  2 
9.  5628656.  2 

9.  9474378 
9. 9607107 
9.5628653 

29. 4710481.  6 
29. 4710295.  6 

29. 4710141.  5 
000.0 

29.  4710137 
0000. 

00.  0000186.  0 

.0=  + 186.0       —    3.0  ({)+ 03.0  (;)— 37.9  (S)+ 37.9  (3)— 27.0  (5)+ 27.0  (§). 
-[-11.0  (i)+ 11.0  (S)-   9.4(1)+   9.4  (3)- 53.7  (?)+ 53.7  (J).] 


Equations  of  condition. 


.0=+  9".619-o=+}-5  +  |-J  +  g 
.0=+15  .394— J  +  f  — 4+ 3  — §+ J 
.0  =  +15   .767-|  +  8-J+§-S  +  § 


Collecting  ter: 
(d) 


I  (rf)  and  dividing  through  by  100  so  as  to  avoid  dealing  with  large  numbers. 


.0=  +1.86+  .507  (5)  +  .030  f  —  .489  (?)  +.379  (|)  — .270  (J). 
+  .176  (§)  +  .110(4)  +  .094  (g)  -.637  (|). 


EBDUCTION  OF  TEIANGULATION. 


71 


Tatle  of  correlatives. 


Direc- 
tion. 

a. 

b. 

0. 

d. 

0/1 
0/2 
0/3 
1/0 
1/2 
1/3 
2/0 
2/1 
2/3 
3/0 
3/1 
3/2 

-1 

■"'+i"' 
+1 

""-i" 

—1 
+1 

"'+i"' 

—1 

'""-i" 

+1 

+.507 
—.489 

+  .176 

+.110 
—.270 

-i 

+1 

+1 

"■'lli 

-1 

+.030 
+  .094 

—1 
+1 

—.537 
+.379 

+1 

Forming  the  normal  eqaations  in  tbe  uaual  manner,  Tve  have : 


0=+  9.619 
0=+15.  394 
0=+15.  767 
0=—  1.860 

+6.  000 
+2.  000 
+2.  000 
—0.  598 

+2.  000 
+6.  000 
—2.  000 
—1.076 

+2.000 
-2.  000 
+6.  000 
+0.  950 

-0.598 
-1.  076 
+  0.950 
+1.  054 

J  find  tlie  following  valnea ; 

a  =  +  1. 900 

6  =  —  4.  386 

c  =  -  5.  208 

d  =  +  3.  059 


Substituting  tlie  values  of  a,  h,  o,  d,  in  the  table  of   correlatives. 


Direction. 

A. 

B. 

C. 

D. 

Correction 

to  each 
direction. 

? 

1 
i 

1 

i 

1 
1 

—1.900 

+4.  386 
-^.386 

4-1.  551 
—1.496 
+0.  538 

+4.037 
—0.674 
—2.770 
—2.486 
+4.  722 
—2.  726 
—0.  822 
-4.294 
+5. 496 
+3.308 
+0.  257 
—4.049 

+5.208 
—5.208 

+1.900 
+1.  900 

—4.386 
+4.386 

+0.336' 
—0.826 

—1. 900 

+4.386 
—4.386 

—5.208 

+0.  092 
+0.  288 

+5.  208 
+5.  208 

—1.  900 
+1.  900 

—1.643 
+1. 159 

—5.  208 

3.0.1 
0.1.3 
1.3.0 

0.1.2 
1.2.0 
2.0.1 

1.2.3 
2.3.1 
3.1.2 

2.3.0 
3.0.2 
0.2.3 

Obsei 

ved 

angles. 

Corrections. 

Corrected  spheri- 
cal angles. 

Sph.  ex- 
cess. 

Plane  a 

agles. 

120 
21 
37 

81 
62 
35 

91 

28 
60 

65 
84 
29 

39 
26 
54 

52 

22 
44 

28 
05 
26 

^■9 

54 
05 

14.  781 
17. 806 
37.180 

51.  222 
38.  500 
45.861 

38.  000 
10.  360 
33.  416 

47.  540 
28.  920 
59.  500 

—3. 308—2. 486. 
—4.037+0.257 
+2.  726—2.  770 

^t.  037-^.  294 
— t.  722—0.  674 
+0.822—2.486 

—4. 722—4.  049 
—5. 496—2. 726 
—0.257—4.294 

—5.496—2.770 
—3.  308—0. 822 
-1-0.  674—4.  049 

120 
21 
37 

39 

26 
54 

08.986 
14.  026 
37. 136 

—  049 

—  049 
—.050 

120 
21 
37 

39 
26 
54 

08.94 
13.98 
37.08 

180 

00 

00. 148 

—.148 

180 

00 

00.00 

81 
62 
35 

44 

42.891 
33. 104 
44.194 

—.063 
—.063 
—.063 

81 
62 
35 

52 
22 
44 

42.83 
33.04 
44.13 

180 

UU 

00. 189 

—.189 

180 

00 

00.00 

91 
28 
60 

28 
05 
26 

29.  229 
02.  138 
28.  865 

—.078 
—.078 
—.078 

91 
28 
60 

28 
05 
26 

29.15 
02.06 
28.79 

180 

00 

00.  232 

—.234 

180 

00 

00.00 

65 
84 
29 

59 
54 
05 

39.  274 
24.  794 
56. 125 

—.064 
—.004 

j     —  065 

65 
84 
29 

59 
54 
05 

39.21 
24.73 
56.06 

181 

00 

00. 193 

—.193 

180 

00 

00.  OO 

72 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


For  a  full  discussion  of  the  Metliod  of  Least  Squares  and  its  application 
to  triang'ulation  see  "A  Treatise  on  the  Adjustment  of  Observations,  by  T. 
W.  Wright,  B.  A.,"  pp.  250-370.     New  York.     D.  Van  Nostrand.     1884. 

COMPUTATION   OF   DISTANCES. 

In  each  triangle,  starting  with  the  base  line,  there  is  known  at  least 
one  side  and  the  three  angles.  The  remaining  sides  are  computed  by  the 
well-known  proportion  of  sides  to  sines  of  opposite  angles,  or  expressed 
h  sin  A 


mathematically,    a  = 


sinB 


In  this    computation  distances  should  be 


used  in  meters,  and  seven  place  logarithms  should  be  employed. 

The  following  is  an  example  of  the  correction  of  the  angles  and  the 
computation  of  the  sides  of  triangles  taken  from  the  work  in  Kansas: 


station. 

Angles  locally 

ad],  and  re- 
duced to  center. 

i  error. 

Piano  angles. 

Log  sines. 

36    29    04.0 
63    58     56.2 
79    31     58.1 

+  .5 
+  .6 
+  .B 

36    29    04.5 
63     58     66.8 
79    31    58.7 

0.2257704 
9.9535952 
9.9927124 

179    59    58.3 
Error=— 1.7 

Log  di3t.  "ffewt- Walton -■ 3.57716H 

Log  sin  Newt 9.9535952 

a.  c.  log  sin  Township  corner 0.2257704 

Log  dist.  Township  comer— '^^llton 3.7565267 

Log  dist.  Newt-Walton 3.57716U 

Log  sin  Walton 9.9927124 

a.  c.  log  sin  township  corner 0.2257704 

Log  dist.  Township  comer — Newt 3.7950439 


COMPUTATION  OF  GEODETIC  COORDINATES. 

The  next  step  is  the  computation  of  the  latitude  and  longitude  of  the 
stations  and  the  azimuth  or  direction  of  the  lines  connecting  them.  Initially, 
the  latitude  and  longitude  of  some  point  is  determined  by  astronomical 
observations,  and  this  point  is  connected  with  the  triangulation.  The 
azimuth,  or  angle  with  a  south  line,  of  a  line  connecting  this  point  with  some 
station  in  the  triangulation  is  also  determined  by  astronomical  observations. 
These,  with  the  observed  angles  and  the  computed  distances  between  the 
stations,  form  the  data  from  which  the  latitudes  and  longitudes  of  the  sta- 
tions and  the  azimuths  of  the  lines  connecting  them  are  computed.     The 


EEDUCTIOISr  OF  TKIANGULATION.  73 

difference  in  latitude  between  two  adjoining-  stations  is  obtained  from  the 
following  equation,  based  upon  the  Clarke  spheroid : 

-dL  =  K  cos  «'  B+K^  sin^  a'  C  +  (dL)  ^D  -  hW"  sin^  a'  E,  '' 

in  which 

c?L  is  tlie  difference  in  latitude. 

K,  the  distance  between  the  stations  in  meters. 

a',  the  fore  azimuth  of  the  line  connecting  them,  measured  round  clock- 
wise from  the  south  through  the  west. 

h,  the  first  term. 

Sh,  the  approximate  difference  in  latitude,  being  the  sum  of  the  first 
twx5  terms. 

B,  C,  D,  and  E,  constants  derived  from  the  dimensions  and  figure  of 
the  earth. 

These  are  given  for  various  latitudes  in  tables  at  the  close  of  the 
volume. 

The  difference  in  longitude  is  obtained  by  means  of  the  following 
formula : 

,, ,      K  sin  a'  A' 

dM=   jr~, — 

cos  L 
in  which 

dM  is  the  difference  in  longitude. 

L',  the  newly  determined  latitude. 

A',  a  constant,  from  tables  near  the  end  of  the  volume,  and  the  others 
as  above. 

The  azimuths  at  the  two  ends  of  a  line  differ  from  one  another,  on 
account  of  the  converg-ence  of  the  meridians.  That  first  determined  is  known 
as  the  fore  azimuth,  the  other,  the  back  azimuth.  All  azimuths  are  meas- 
ured from  the  south  jDoint  around  to  the  right. 

The  back  azimuth  is  computed  from  the  formula: 

sin  (L+L^ 


•  da  ^=  dM 


cos  ^  dl^ 

where  M  is  the  longitude  of  the  first  station. 
L,  the  latitude,  and 
L'  the  latitude  of  the  second  station. 


74 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


The  constants  used  are  those  of  the  Clarke  spheroid  of  1866. 

These  formulae  are  derived  and  explained  in  Appendix  No.  7,  Report 
U.  S.  Coast  and  Geodetic  Survey  for  1884. 

The  following  are  examples  of  the  use  of  the  formulae,  taken  from  the 
ti-i anovulation  in  New  Mexico  : 


Spherical  angle: 

Azimuth  a' : 

6  .1  +  180° 


Nell— Chuaca. 
Nell— Zuui. 
Znui— Nell. 


159 

120 

29 
54 

08. 728 
13.  980 

38 
179 

34 
50 

54.  748 
02.  124 

218    24     56.872 


Geodetic  Cooedinates. 


LONGITUDE. 


35    07    25.927 


log.K 


4. 6236305 
8.  5111933 
COS  ,V  9.  8930500 


log  (I)        3. 0278738 

log.  K2  9.  24726 

"     C  1.25696 

"    sin^  a'    9. 58986 

log.  (II)         0. 09408 

log.  D  2.  3679 

'■     [I+II]'   6.0568 


A'  108     54    40.285 

Computation  for  longitude : 

log.  K  4.  6236305 

"    sin  a'  9.7949286 

"A'  8. 5092394 

"    sec.L'  0.0872944 

rr.  for  diff.  arc.  &  sine  =  —  15 


log.  (V) 


Computation  of  azimuth : 


log.  E  6.  0124 

"    K2sin2a'8.8371 
"     (I)  3. 0279 

log.  (TV)         7. 8774 


2. 776614 

-  597".  S76 

-  9'     57". 876 


Azimuth  check. 


(I) 

(ID 

1066.  286+ 
1.242  + 

.026+ 
.008- 

(ni) 

(IV) 

log.      "    "         3.0283792   Check: 
"[I+II)=       6.0567584   Spher.  angle 
at 

-«L 

1067.546+ 

* 

Computation  of  Azimuth,  a,  in  Book  ,  i 

Splierical  angle  and  distance  =  K,  in  Book 
Station;  Computed  by 


,  page,  Triangle  No. 


Azimntli  a: 
Spherical  angle : 


Chusca — Nell. 

339 
25 

21 
11 

40. 150 
38.  601 

Chusca — Zuni. 

4 
179 

33 

57 

18.  751 
25.  650 

Zuni-Cliuaca. 

184 

30 

44.  401 

PEIMAEY  TEAVEESES. 


75 


Geodetic  Coordinates. 


LONGITUDE. 


35     07    25. 928 


log.  K  4. 9280539 

"     B  8. 5111594 

"    COS  a'  9. 


log.  (I)         3. 4378393 


log.  K?  9. 85610 

'P  C  1.26435 

"    siu'a'  7.79982 

log.  (II)  8.  92027 

log.  D  2. 3703 

•■'  [I+n]'^  6.  8757 

log.  (Ill)  9.2460 

log. E  6. 0214 
"    K^sin' a' 7.6.559 

■'    (I)  3.4378 

log.  (IV)  7.1151 

(I)  2740."560+  I 

(II)  .083+ 


Computatii 


T.  fordiff.i 
.(V) 


108    50    14.518 
+  4    25.  768 

108    54    40. 286 

I  forlongilude: 

4.  9280539 


8. 1.092394 

0. 0872944 

c&sine   -129 

2. 4245028 
+265".  761 


Computation  of  azimut  Ii : 


.(H^\ 


log.  (TI) 


2. 424503 
9. 764002 


2. 188514 
■      154".  350 
•2'     34".  350 


A  zlmutti  check : 


— 6L     +2740.818 


.  176+  [I+II]  2740.  643 

.001-  !  log.      •■  3.4378525 

[I+IIJ2    6.875705 


Check : 
Spher.  anj 
at  Zuni 


33     54     12. 471 
33     54     12. 469 


Computation  of  Azimuth  a,  in  Book  67,  page  4. 

Spherical  angle  and  distance  =  K,  in  Book  64,  page  12,  Iriangle  No.  3. 
Station;  Computed  by  H.  M.  W. 

When  the  hnes  are  not  more  than  twenty  miles  in  length,  the  equation 
for  latitude  may  be  simplified  without  appreciable  error  by  di'opping  the 
last  two  terms. 

TRAVERSE  LINES  FOR  PRIMARY  CONTROL. 

In  level  country,  especially  if  it  is  covered  with  forests,  it  is  very  expen- 
sive to  carry  on  triangulation,  and  in  some  cases  practically  impossible  to 
do  so.  Under  such  circumstances  the  only  means  of  obtaining  an  adequate 
control  for  maps  is  by  means  of  traverse  lines. 

A  traverse  line  consists  of  a  series  of  direction  and  distance  measure- 
ments. Each  course,  as  the  du-ection  and  the  accompanying  distance  are 
called,  depends  upon  the  one  immediately  preceding  it,  and  a  continuous 
chain  is  thus  formed.  Traverse  lines  are  largely  used  in  the  topographic 
work  proper  for  making  minor  locations.  The  primary  traverse  diifers 
from  these  only  in  the  fact  that  it  is  much  more  elaborately  executed. 

The  initial  point  of  a  primary  traverse  must  be  located  either  by 
triangulation  or  by  astronomic  determinations.     The  end  of  the  line  should, 


76  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

if  possible,  be  a  point  similarlv  well  located.  The  line  sliould,  if  practica- 
ble, follow  a  railroad,  in  order  to  obtain  the  easiest  possible  grades,  and 
thus  avoid  errors  incident  to  slope. 

The  instrument  used  for  measuring-  directions  should  have  a  circle  6  to 
8  inches  in  diameter,  and  should  read  by  vernier  to  10  seconds.  The  the- 
odolites formerly  used  in  the  primary  triaugulation  are  generally  used  in 
this  work.  A  larger  or  more  elaborate  instrument  is  not  advisable  on 
account  of  the  difficulties  of  transporting  it  and  frequently  setting  it  up. 
Upon  short  lines  instruments  reading  to  minutes  may  be  used. 

The  readings  should  be  upon  signals  consisting  of  poles,  and  fore  and 
back  rodmen  must  be  employed  for  carrjnng  and  setting  them.  The 
angular  measurements  between  the  poles  should  be  read  by  both  verniers, 
and  it  is  advisable  to  note  the  compass  readings  at  the  same  time,  in  order 
to  avoid  gross  errors.  At  intervals  of  10  to  20  miles,  depending  upon  the 
number  of  courses  to  a  mile,  observations  should  be  made  for  azimuth, 
obsei*ving  for  this  purpose  upoii  the  pole  star,  preferably  at  elongation. 

The  measurements  of  distance  are  effected  by  the  use  of  steel  tapes, 
and  preferably  by  300-feet  tapes,  similar  to  those  used  in  measuring  base 
lines.  Two  chainmen  should  be  employed,  and  in  order  to  avoid  eri'ors  in 
the  count,  it  is  well  to  count  the  rails,  in  case  the  woi-k  is  done  upon  rail- 
road tracks. 

The  temperature  should  be  noted  by  means  of  thermometers  at  frequent 
intervals,  in  order  that  the  proper  corrections  may  be  applied. 

The  errors  incident  to  running  primary  traverses  are  of  two  classes: 
errors  of  direction  and  errors  of  distance. 

Those  of  direction  are  similar  to  those  treated  of  under  the  head  of 
Instructions  for  the  Measurement  of  Horizontal  Angles,  and  need  not  be 
specified  here. 

Owing  to  the  necessity  of  setting  up  the  theodolite  at  frequent  inter- 
vals, it  is  impracticable  to  observe  at  each  station  the  series  of  angles  speci- 
fied in  the  abdve-mentioned  instructions,  and  only  a  single  or  at  the  most 
a  double  measure  of  the  included  angle,  with  the  reading  of  each  vernier, 
is  practicable  for  the  measurement  of  direction.  It  is  here  provided  that 
observations  for  azimuth  upon  Polaris  should  be  much  more  frequent  than 
in  triangulation,  and  thus  an  absolute  correction  to  the  dii-ections  is  intro- 


ELEVATIONS.  •  77 

duced  mucli  ofteuer.  At  each  azimuth  station  the  new  astronomic  azimuth 
should  be  adopted  in  place  of  that  carried  forward,  and  in  case  the  discrep- 
ancy between  the  two  is  sufficiently  great  to  involve  perceptible  error  upon 
the  scale  of  the  map,  the  correction  should  be  uniformly  distributed  forward 
from  the  first  station. 

In  running  these  traverses  all  road  crossings  should  be  located,  as 
topographic  traverses  will  be  run  over  the  roads  and  will  be  connected  with 
the  primary  traverses  at  these  points.  All  prominent  houses  or  natural 
features  of  any  kind  in  sight  from  the  line  must  be  located  by  iatersection, 
as  they  will  doubtless  be  used  by  the  topographers  for  location. 

When  traversing  in  a  country  which  has  been  surveyed  by  the  Greneral 
Land  Office  into  townships  and  sections,  the  crossing  of  every  township 
and  section  line  should  be  located,  and  the  directions  of  the  township  lines 
with  reference  to  the  line  of  traverse  should  be  carefully  measured  in  order 
to  establish  as  close  a  relation  as  possible  between  the  traverse  line  which 
serves  as  ultimate  control,  and  the  township  system  of  surveys,  which  serves 
as  a  secondary  control. 

Lines  of  traverse  exceeding  100  miles  in  length  should  be  reduced  by 
computation.  The  distances  should  be  corrected  for  error  of  tape,  for  tem- 
peratiTre,  and  slope,  and  should  be  reduced  to  sea  level,  in  the  same  man- 
ner as  above  described  in  treating  of  the  reduction  of  base  lines,  in  case 
these  corrections  are  of  sufficient  amount  to  affect  the  length  appreciably 
upon  the  map. 

The  courses  should  be  corrected  for  convergence  of  meridians.  Then, 
commencing  at  the  initial  point,  the  latitude  and  departure  of  each  station, 
one  from  another,  should  be  computed  in  feet.  The  sum  of  the  latitudes 
converted  into  seconds  of  latitude  gives  the  difference  in  latitude,  and  the 
sum  of  the  departures  converted  into  seconds  of  longitude  gives  the  differ- 
ence in  longitude. 

Short  lines  of  traverse  may  be  platted  with  minute  reading  protractors, 
but  in  this  platting  the  utmost  care  should  be  exercised. 

PRIMARY   ELEVATIONS. 

The  initial  elevations  of  this  work  are  derived  from  various  sources. 
Any  trustworthy  results  known  to  be  of  a  sufficient  degree  of  accuracy  for 


78  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

the  2:)urpose  may  be  adopted.  Whenever  elevations  have  been  determined 
within  the  area  to  be  surveyed  by  the  United  States  Coast  and  Geodetic 
Siu-vey  or  the  United  States  Lake  Survey,  they  may  be  accepted  without 
question.  The  work  of  these  organizations  has  been  sketched  in  the  early 
pai-t  of  this  volume  and  is  shown  upon  map  No.  1. 

Wlien  these  determinations  are  not  available,  initial  bench  marks 
should,  if  possible,  be  obtained  from  the  profiles  of  railroads  traversing 
the  district.  These  have  been  adjusted  and  the  results  published  in  the 
Dictionary  of  Altitudes  (Bulletin  No.  76,  U.  S.  Geological  Survey).  In 
case  there  are  no  raih'oads  to  furnish  initial  datum  points,  as  may  occur 
in  the  sparsely  settled  regions  of  the  West,  or  the  profiles  available  are 
regarded  as  untrustworthy,  it  may  become  necessary  to  use  barometric 
observations.  Where  a  series  of  these,  of  a  year  or  more  in  length  is 
available,  the  result  may  be  regarded  as  sufficiently  trustworthy  for  this 
pvu-pose. 

In  regions  where  secondary  triangulation  is  practicable  the  measure- 
ment of  heights  may  be  taken  up  with  the  plane  table  directly  from  datum 
points,  as  above  indicated,  and  carried  throughout  the  work  by  means  of 
this  instrument.  Otherwise  it  becomes  necessary  to  do  more  or  less  level- 
ing in  order  to  extend  and  multiply  datum  points  to  control  the  less 
accurate  work  connected  with  the  traversing.  If  practicable,  the  wye 
level  should  be  employed. 

The  extent  of  the  work  of  the  wye  level  which  may  be  required 
depends  mainly  upon  the  contour  interval  of  the  map  to  be  made.  It  may 
be  said  in  general,  that  a  single  line  aci-oss  a  sheet  will  furnish  a  sufficient 
number  and  a  suitable  distribution  of  points  for  the  proper  correction  of 
the  subsequent  work.  Wherever  practicable  such  lines  should  be  run 
along  raih'oads,  in  order  to  obtain  easy  grades  and  thus  lig-hten  the  work. 
When  railroads  are  not  available,  they  should  be  run  along  wagon  roads, 
selecting,  so  far  as  they  will  suit  the  purpose,  those  having  the  easiest 
grades  and  the  straightest  com-ses. 

Where  the  control  of  the  map  is  effected  by  means  of  primary  ti-av- 
ersing,  such  traverse  should  be  accompanied  by  a  level  line,  unless  that  of 
the  raih'oad  which  the  traverse  follows  appears  to  be  of  sufficient  accuracy. 


CHAPTER    IV. 

SECONDARY   TRIANGULATION. 

The  work  of  making  secondary  locations  by  intersection  is  done  mainly 
by  plane  table.  The  use  of  the  theodolite  for  this  purpose  is  restricted  to 
those  cases  where  but  little  of  this  kind  of  location  can  be  effected,  and 
where,  therefore,  it  seems  scarcely  worth  while  to  prepare  plane-table  sheets. 

By  means  of  the  primary  triangulation,  two  or  three  points  are  usually 
located  upon  each  atlas  sheet.  Within  this  primary  triangulation,  and 
depending  upon  it,  are  then  located  a  large  number  of  points,  either  by 
intersection,  by  traverse,  or  by  both  methods,  forming  a  geometric  frame- 
work upon  which  the  sketching  of  the  map  depends. 

Location  by  intersection  should  be  carried  as  far  as  practicable — that  is, 
all  points  capable  of  being  located  in  this  manner  should  be  so  located  in 
order  to  afford  the  most  ample  control  possible  for  the  traverse  hues,  by 
which  the  intervening  areas  are  to  be  filled  in,  it  being  understood  that  the 
location  by  intersection  is  more  accurate  and  more  rapid,  and  consequently 
in  every  way  more  economic,  than  location  by  traverse. 

THE    PLANE    TABLE. 

Much  misapprehension  exists,  especially  in  this  country,  regarding  the 
character  and  application  of  this  instrument.  This  arises,  apparently,  from 
the  fact  that  it  is  little  known.  For  making  a  map  the  plane  table  is  a  uni- 
versal instrument.  It  is  appHcable  to  all  kinds  of  country,  to  all  methods 
of  work,  and  to  all  scales.  For  making  a  map  it  is  the  most  simple,  direct, 
and  economic  instrument;  its  use  renders  possible  the  making  of  the  map 
directly  from  the  country  as  copy,  and  renders  unnecessary  the  making  of 
elaborate  notes,  sketches,  photographs,  etc.,  which  is  not  only  more  expen- 
sive, but  produces  inferior  results. 


yO  A  MANUAL  OF  TOrOGEAPHlO  METHODS. 

The  plane  table  is  essentially  very  simple,  consisting'  of  a  board  upon 
which  is  fastened  a  sheet  of  drawing  paper.  This  board  is  mounted  upon 
a  tripod,  which,  in  the  more  elaborate  forms  of  the  instrument,  possesses 
great  stiffness  and  stability.  It  should  be  capable  of  being  leveled,  of 
being  tm-ned  in  azimuth,  and  of  being  clamped  in  any  position.  Upon  the 
paper  is  produced  directly  in  miniature  a  representation  of  the  country. 
When  set  up  at  various  places  within  the  area  in  process  of  being  mapped^ 
the  edges  of  the  board  must  always  be  placed  parallel  to  themselves — that 
is,  a  certain  edge  of  the  board  must  always  be  set  at  the  same  angle  with 
the  north  and  south  line.     This  is  called  orienting  the  board. 

Directions  are  not  read  off  in  degrees  and  minutes,  but  platted  directly 
upon  the  paper.  The  instrument  used  for  this  purpose  is  known  as  the 
alidade,  and  consists  of  a  ruler  with  a  beveled  edge,  to  which  are  attached 
for  i-ough  work  two  raised  sights,  and  for  the  higher  class  of  work  a  tele- 
scope-turning on  a  horizontal  axis.  This  telescope  carries  also  a  delicate 
level  and  a  vertical  arc  for  the  measurement  of  angles  in  the  vertical  plane, 
from  which  relative  heights  are  obtained.  The  method  of  using  this  instru- 
ment is  extremely  simple  in  principle,  and  becomes  difficult  in  practice  only 
when  a  high  degree  of  accuracy  is  required. 

The  work  of  making  locations  from  intersections  obtained  by  means  of 
the  plane  table  requires  that  the  instrument  have  the  utmost  stability  con- 
sistent with  lightness  and  portability.  It  requires  an  alidade  equipped  with 
a  telescope  of  considerable  power  and  good  definition.  In  short,  it  requires 
that  the  plane  table  be  in  every  respect  of  the  best  modern  type  in  order 
that  the  highest  degree  of  accuracy  possible  to  represent  upon  the  paper  be 
attained.  Various  forms  of  plane-table  movement  have  been  in  use,  includ- 
ing the  heavy  and  cumbersome  but  stable  movement  of  the  Coast  and  Geo- 
detic Survey,  and  the  light  but  unstable  movement  used  by  the  same 
organization  in  its  less  important  work.  At  present  a  table  is  in  general  use 
which  was  invented  by  Mr.  W.  D.  Johnson,  of  this  Survey,  which  combines 
the  elements  of  stability,  lightness,  and  facility  of  operation  in  a  remarkable 
(leo-ree.  (See  Fig.  8.)  This  movement  is  essentially  an  adaptation  of  the 
ball-and-socket  principle,  so  made  as  to  furnish  the  largest  practicable 
amount  of  bearing  surface.     It  consists  of  two  cups,  one  set  inside  the  other. 


JOHNSON    PLANE  TABLE  AND  TELESCOPIC  ALIDADE, 


THE  PLANE-TABLE. 


81 


the  inner  surface  of  one  and  the  outer  surface  of  the  other  being  ground  so 
as  to  fit  accurately  to  one  another.  The  inner  cup  is  in  two  parts,  or  rather 
consists  of  two  rings  one  outside  the  other,  the  one  controlling  the  move- 
ment in  level  and  the  other  that  in  azimuth.  From  each  of  these  rings  there 
projects  beneath  the  movement  a  screw,  and  upon  each  of  these  screws  is  a 
nut  by  which  it  is  clamped.     There  is  no  tangent  screw  for  either  the  leveling 


Johnson  Plan  e-IAble  Head 


a. Plana  Table  board  f.  VpperLepel  Cup 

b^  Bearing  PLaze  g.  Ztofr'er     " 

<?.  TripocLHeaa  fi..  Jjei'sL  Clamp 

el.       •'      Z,effs  t   Mzimzith,  CLamp 
e.jiziTruztfiGzp 


Fig.  b.— Joliuson  plane-table  tripod  liead.    Section. 

or  the  azimuth  motion,  as  none  is  required.  The  movement  is  sustained 
by  a  light  hard-wood  tripod  with  split  legs.  The  board  used  generally 
accommodates  a  full  atlas  sheet,  but  necessarily  differs  in  size,  owing  to 
the  different  scales  of  field  work  adopted.  The  largest  board  used  for 
this  movement  holds  an  atlas  sheet  upon  a  scale  of  1:45000,  and  is  24  by  36 
inches  in  size. 

MON  SXII 6 


82  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

The  question  of  paper  for  the  ^Diane-table  sheets,  especially  in  inter- 
section work,  is  of  great  importance,  as  paper  which  expands  and  contracts 
differently  in  different  directions  under  varying  conditions  of  moisture  will 
easily  produce  errors  of  magnitude  in  the  work.  It  matters  little  if  the 
paper  contracts  and  expands,  provided  it  does  so  uniformly  in  all  dnections, 
but  all  paper  is  made  with  more  or  less  fiber,  and  accordingly  expands  and 
contracts  more  in  one  direction  than  in  another.  To  cotmteract  this,  two 
thicknesses  of  paper  are  used,  preferably  that  known  as  Paragon  paper, 
moxmted  with  .  the  grain  of  the  two  sheets  at  right  angles  to  one  another, 
and  with  cloth  between  the  la5'ers.  In  sheets  so  prepared  it  has  been  found 
that  there  is  practically  no  distortion,  even  under  the  most  severe  tests. 

The  board  is  generally  made  of  seasoned  white  pine,  from  one-half  to 
five-eighths  of  an  inch  thick,  with  cleats  across  the  ends  fastened  in  such  a 
way  as  to  allow  the  body  of  the  board  to  contract  and  expand  freely,  and 
therefore  without  warping.  Into  the  corners  of  this  board  and  on  the  edges 
at  points  halfway  between  the  corners  are  set  female  screws  for  holding 
the  paper  to  the  board.  At  corresponding  points  in  the  plane-table  sheet 
are  punched  holes  half  an  inch  in  diameter  which  are  lined  with  eyelets, 
and  thi'ough  which  pass  screws  with  broad  heads  fitting  into  the  female 
screws  in  the  boai-d.  The  holes  in  the  paper,  being  larger  than  the  screws, 
allow  the  paper  to  expand  or  contract  freely  when  the  screws  are  loose. 
When  tightened,  the  broad  heads  of  the  screws  bind  the  paper  firmly  in  place. 

THE    ALIDADE. 

The  ahdade  used  with  this  plane  table  consists  of  a  ruler  of  brass  or 
steel  18  inches  to  2  feet  in  length,  graduated  upon  a  chamfered  edge  to  suit 
the  scale  of  work,  and  carrying  upon  a  column  a  telescope  having  a  focal 
distance  of  12  to  15  inches  and  a  power  of  about  15  diameters.  It  has  a 
vertical  arc  reading  by  vernier  to  single  minutes,  and  a  delicate  level  upon  the 
telescope.  In  some  alidades  there  is  an  adjustment  to  make  the  zeros  of  the 
vertical  arc  and  the  veiTuer  coincide,  when  the  telescope  is  horizontal,  while 
in  others  it  is  necessary  to  read  the  index  error  of  the  vertical  arc  and  correct 
for  it,  there  being  no  such  adjustment.  The  telescope  turns  in  a  sleeve,  for 
adjustment  of  vertical  collimation. 


THE  PLANE-TABLE.  83 

Upon  the  j)lane-table  sheet  is  constructed  a  projection  upon  the  scale 
of  the  field  work,  and  upon  that  are  platted  such  of  the  primary  points  as 
fall  upon  the  sheet,  each  plane  table  sheet  being  made  to  correspond  to  an 
atla,s  sheet.     These  primary  points  are  first  occupied  by  the  plane  tabler. 

The  instrument  is  set  over  one  of  these  stations,  leveled,  and  clamjjed. 
The  ruler  edge  of  the  alidade  is  then  laid  upon  the  line  connecting  this 
station  with  a  neighboring  one  upon  the  sheet,  and  the  table  turned  until 
the  other  station  is  upon  the  vertical  wire  in  the  telescope.  The  instrument  is 
then  oriented,  and,  after  clamping  in  azimuth,  is  ready  for  work.  Keeping 
the  ruler  upon  the  occupied  station  on  the  sheet,  the  telescope  is  then  turned 
upon  other  objects  which  it  is  desirable  to  locate,  and  lines  are  drawn,  in 
turn,  toward  them.  The  instrument  is  then  taken  up  fCnd  moved  to  a  second 
station,  where  it  is  again  set  up,  leveled,  and  oriented,  as  before.  A  sight  is 
then  taken,  and  a  line  drawn  in  the  direction  of  each  point  sighted  from  the 
first  station,  and  the  intersection  of  each  pair  of  sight  lines  is  the  true  position 
of  the  corresponding  point  upon  the  map.  In  this  way,  station  after  station 
is  occupied  by  the  plane  table,  and  numerous  points  are  located  by  inter- 
section. If  possible,  each  point  thus  located  should  be  intersected  from  at 
least  three  stations  in  order  to  verify  its  location. 

Any  point  thus  located  on  the  map  may  be  used  afterward  as  a  station. 
In  case  it  is  necessary  to  occupy  a  point  toward  which  no  line  has  been 
drawn,  or  which  has  not  been  located,  the  simplest  and  best  plan  for  effect- 
ing its  location  is  as  follows : 

Fasten  upon  the  plane-table  board,  which  necessarily  has  not  yet  been 
oriented,  a  piece  of  tracing  linen,  or  ,in  default  of  that,  a  piece  of  tracing 
paper.  Assume  a  point  upon  this  linen  to  represent  the  station,  take  sights 
upon,  and  draw  lines  to  all  located  points  within  the  range  of  vision,  and 
then,  loosening  the  linen  from  the  board,  move  it  about  over  the  map  until 
these  sight  lines  fall  upon  the  proper  points  upon  the  map.  Then  prick 
through  the  position  of  the  station  from  the  linen  to  the  map  underneath. 
This  location  should  then  be  tested  by  sighting  from  the  point  thus  found 
to  the  various  objects  to  see  if  the  sight  lines  fall  upon  the  points  as  marked 
upon  the  map. 


84  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

In  case  one  line  of  sig'lit  upon  the  required  station  has  been  obtained, 
that  sight  Hue  may  be  utihzed  in  making  the  location  as  follows  by  resec- 
tion: Having  leveled  the  table,  place  the  alidade  upon  this  sight  line 
already  drawn,  with  the  telescope  pointing  toward  the  object  from  which 
the  sight  was  taken.  Then  turn  the  table  in  azimuth  until  the  telescope 
falls  iipon  this  point,  and  clamp  it.  The  table  is  now  oriented,  but  the  posi- 
tion of  the  present  station  is  unknown  further  than  that  it  is  known  to  be  upon 
this  line.  Then  select  some  station  whose  direction  makes  a  wide  angle 
with  this  line,  and  move  the  alidade  until  the  cross  wire  falls  upon  this 
selected  station,  while  the  ruler  at  the  same  time  is  upon  the  representation 
of  the  station  upon  the  map.  The  ruler  will  then  cross  the  sight  line  at 
the  point  desired.  By  way  of  check,  repeat  the  process  with  another  sta- 
tion or  located  point.  For  this  purpose  a  point  in  suitable  direction  is 
valuable  in  proportion  to  its  proximity. 

Using  the  instrument  as  described  above,  the  topographer  locates  from 
*them  all  possible  points.  Then  visiting  in  turn  such  of  them  as  he  finds 
necessaiy,  pei'haps  a  dozen  or  twenty,  he  locates  by  intersection  points  all 
over  the  sheet  in  as  great  number  and  as  well  distributed  as  possible,  and 
with  special  reference  to  the  needs  of  the  traverse  men,  who  will  come  after 
him  and  whose  work  will  be  located  by  means  of  his  determinations.  All 
this  work  must  be  done  with  the  utmost  nicety  and  precision.  The  setting 
of  the  alidade  upon  the  station  must  bisect  the  needle  hole  by  which  it  is 
marked  and  the  lines  of  direction  must  be  drawn  with  a  sharp-pointed  pencil. 

The  necessity  for  precision  will  be  recognized  when  it  is  understood 
that  any  error  introduced  in  the  early  part  of  the  plane-table  triangulation 
will  be  not  only  perpetuated,  but  increased  many  times  over  as  the  work 
progresses,  and  as  soon  as  an  error  becomes  appreciable  it  produces  difficul- 
ties and  uncertainties  in  making  locations,  which  may  lead  to  embarrassing 
delays,  and  ultimately  require  that  all  the  work  be  repeated. 

MEASUREMENT   OF   ALTITUDES. 

While  making  horizontal  locations  of  points  with  the  plane  table,  their 
heights  must  also  be  measured,  relative  to  that  of  the  point  occupied.  This 
is  done  by  means  of  the  vertical  arc  of  the  alidade  and  the  level  upon  the 


TEAVBESE  WOEK.  85 

telescope.  Pointing  upon  tlie  object  whose  relative  height  is  to  be  measured, 
the  telescope  must  first  be  brought  to  a  horizontal  position.  In  case  the 
vertical  arc  is  movable,  its  zero  must  then  be  brought  to  the  zero  of  the  ver- 
nier. In  case  it  is  not  movable,  the  index  error,  with  its  sign,  must  be  read. 
The  telescope  is  then  raised  or  depressed  to  the  point  and  the  reading 
obtained.  This  adjustment  of  the  vertical  arc  or  reading  of  the  index  error 
must  be  done  for  each  point,  as  the  table  cannot  be  leveled  with  sufficient 
accuracy,  or  cannot  be  expected  to  maintain  its  level,  so  as  to  dispense  with 
it.  Knowing  the  horizontal  distance  to  the  point  and  the  angle  of  elevation 
and  depression,  the  difterence  in  height  is  obtained  by  the  solution  of  a  right- 
angled  triangle,  thus: 

h  =:d  tang  a, 
Ji  being  the  difference  in  height,  c?  the  distance,  and  a  the  angle  of  elevation 
or  depression.  This  distance  is  then  to  be  corrected  for  curvature  of  the 
earth  and  for  refraction  by  the  atmosphere.  The  correction  for  the  former 
is  obtained  with  sufficient  accuracy  by  the  following  empirical  rule.  The 
curvature  in  feet  equals  two-thirds  the  square  of  the  distance  in  miles.  It  is 
always  positive  in  sign,  whatever  may  be  the  sign  of  the  difference  in  height. 

Refraction  is  an  uncertain  and  variable  quantity.  It  is  usually  greatest 
at  morning  and  night  and  least  at  midday.  It  is  greater  the  nearer  the  line 
of  sight  is  to  the  ground.  Often  in  desert  regions  it  is  excessive  in  amount. 
It  is  usually  assumed  at  one-seventh  the  curvature,  and  is  negative. 

Tables  for  the  solution  of  vertical  angle  work  are  appended  to  this 
volume.  These  give  differences  in  height  for  all  angles  and  distances  which 
should  be  employed,  with  corrections  for  curvature  and  refraction. 

Differences  of  height  should  not  be  measured  at  greater  distances  than 
10  miles,  if  it  can  be  avoided.  An  error  of  1'  in  the  measurement  of  the 
angle  is  at  this  distance  about  15  feet,  while  the  uncertainty  of  refraction 
in  such  a  length  of  line  is  necessarilj^  great. 


TRAVERSE  WORK. 


As  Stated  above,  under  the  head  of  primary  traverses,  a  traverse  line 
consists  of  a  series  of  direction  and  distance  measurements  depending  upon 
one  another.    These  lines  should  be  connected  wherever  possible  with  trian- 


86  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

gulation  points  iu  order  to  check  up  accumulated  errors.  If  it  were  prac- 
ticable or  econoroic  to  carry  on  all  tlie  work  of  location  by  intersection,  this 
would  be  the  most  accurate  and  on  most  accounts  the  best  way  to  effect  it, 
but  it  is  only  in  limited  localities,  such  as  high  mountain  regions,  where  bold 
topographic  forms  predominate  and  where  there  is  little  or  no  culture,  that 
the  method  of  intersection  is  practicable  for  locating  all  necessary  points.  It 
is  probable  that  in  nine-tenths  of  the  area  of  the  United  States  it  will  be 
found  necessary  to  locate  the  details  of  topography,  culture,  and  di-ainage 
by  means  of  traverse  lines.  In  different  parts  of  the  country  the  relative 
extent  to  which  the  two  methods  can  be  applied  depends  upon  various 
circumstances,  .principally  the  amount  of  relief  of  the  surface  and  the  prev- 
alence of  forests.  Thus  upon  the  Atlantic  Plain,  which  is  densely  covered 
with  forest,  and  which  is  very  level,  it  is  necessary  to  use  the  traverse 
method  exclusively,  including  even  the  primary  control.  Passing  from  this 
as  an  extreme  case,  through  rolling  and  hilly  country  to  the  high  sharp 
mountains  of  the  West,  the  triangulation  method  becomes  more  and  more 
prominent  while  the  traverse  method  finally  becomes  used  but  little,  except 
in  the  details  of  roads  and  other  cultural  features. 

For  executing  traverse  work  various  instruments  have  been  in  use  for 
measuring  both  distances  and  directions.  For  direction  there  have  been 
used  theodolites  of  various  forms  and  prismatic  compasses  and  for  distances 
the  stadia  and  the  wheel. 

At  present  all  traverse  work  is  done  with  plane  tables,  upon  which  the 
directions  and  distances  are  platted  directly.  The  plane  table  used  for  this 
purpose  is  of  the  simplest  possible  form,  consisting  of  a  board  about  16 
inches  square,  into  one  edge  of  which  is  set  a  narrow  box  containing  a  com- 
pass needle  3  inches  in  length.  This  table  is  supported  by  a  tripod  of  light 
construction,  without  leveling  apparatus,  the  leveling  of  the  instrument 
being  effected  with  sufficient  accuracy  by  the  tripod  legs.  A  single  screw 
fastens  the  board  to  the  tripod  head  and  the  adjustment  in  azimuth  is  made 
by  simply  tm-ning  the  board  with  the  hand.  It  is  held  in  place  by  friction. 
The  table  is  adjusted  in  azimuth,  or  oriented,  by  means  of  the  compass 
needle — that  is,  it  is  turned  until  the  needle  rests  opposite  the  zero  marks 
in  the  compass  box,  and  is  thus  always  made  approximately  parallel  to 
itself,  provided  the  magnetic  declination  remains  constant. 


TRAVERSE   PLANE  TABLE  AND   RULER  ALIDADE. 


TRAVERSE  WORK. 


87 


The  alidade  consists  of  a  brass  ruler,  12  inches  long,  with  folding  sights. 
The  edge  of  the  ruler  is  graduated  to  facilitate  platting  of  distances.  Ordi- 
nary drawing  paper  backed  with  cloth  is  used  for  plane-table  sheets,  and  is 
attached  to  the  board  by  thumb  tacks.        * 

When  traversing  is  done  along  roads,  as  is  commonly  the  case,  dis- 
tances are  measured  by  counting  the  revolutions  of  a  wheel,  usually  one 
of  the  front  wheels  of  a  buggy  or  buckboard.  For  counting  the  revolutions, 
various  automatic  devices  have  been  in  use.  The  old  form  of  odometer 
known  as  the  pendukim  was  first  tried  and  was  unqualifiedly  condemned. 
The  form  now  in  general  use  was  devised  by  Mr.  E.  M.  Douglas  of  this 
Survey.     See  Fig.  9. 


^■"'Vi/viaocctf^ 


Fig,  9. — Douglas  odometer. 

For  operating  this  a  cam  is  placed  on  the  hub  of  the  wheel,  which  by 
raising  a  steel  spring  as  the  wheel  revolves  carries  the  index  forward  one 
division  for  each  revolution.  This  form  is  the  most  trustworthy  that  has 
yet  been  devised,  but  is  not  altogether  satisfactory,  and  many  topographers 
prefer  to  count  the  revolutions  of  the  wheel  directly,  using  an  an-angement 
by  which  a  bell  is  rung  at  each  revolution. 

An  experience  covering  many  thousands  of  miles  of  measiu'ement  has 
shown  that  as  a  working  method  of  measuring  distances  on  roads  the  wheel 
is  superior  to  the  stadia,  alike  as  to  accuracy  and  rapidity 

A  traverse  "man  is  generally  assigned  a  tract  of  country  within  which 
he  is  instructed  to  run  traverses  of  all  the  public  roads  and  of  such  of  the 
private  roads  as  appear  to  be  necessary  in  order  to  control  the  entire  tract. 
If  practicable,  he  is  furnished  with  the  positions  of  the  points  located  within 


88  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

liis  ti-act  properlv  platted  upon  Lis  plaue-table  sheet,  or,  if  tliese  cau  uot  be 
t\u-nished,  with  such  descriptions  of  them  as  are  necessary  to  enable  him  to 
recognize  them  and  close  his  lines  npon  them  or  connect  with  them  by  tri- 
angnlation.  He  is  fm-nished  mth  a  horse  and  buggy  or  buckboard,  traverse 
plane  table,  and  aneroid.  He  has  no  rodman,  but  is  expected  to  sight  natural 
objects.  Setting  up  his  instrument  at  his  initial  station,  he  leA^els  it  roughly 
by  means  of  the  tripod  legs,  orients  it  by  turning  the  table  until  the  com- 
pass needle  is  on  the  zero  mark  in  the  compass  box,  then,  marking  a  point 
on  the  paper  to  represent  his  initial  station,  and  placing  his  alidade  upon  it, 
he  points  it  to  an  object  selected  as  his  second  station,  and  draws  a  line  in 
that  direction.  Driving  along  the  road  he  passes  the  point  sighted  at,  noting 
the  distance  to  it  by  the  reading  of  the  odometer,  or  the  coimt  of  the  revolu- 
tions of  the  wheel,  and  the  height  as  recorded  by  the  aneroid,  and  passes  on, 
selecting  some  point  from  which  he  can  see  the  point  sighted  at.  There  he 
stops,  sets  up  his  table  as  before,  orients  it,  and  sights  upon  the  same  signal 
which  he  sighted  fi-om  his  initial  station.  He  plots  the  distance  to  the  signal 
along  the  sight  line  from  his  initial  station;  then  from  the  location  of  the  sig- 
nal as  thus  established  he  plots  liis  second  station  by  the  distance  measure- 
ment and  the  reverse  of  the  observed  direction.  In  this  way  the  work  pro- 
gresses, a  hundred  stations  or  more  being  occupied  in  the  course  of  the  day. 
In  this  work  one  should  aim  to  make  as  few  stations  and  to  take  as  long 
sights  as  possible  consistent  with  accuracy.  Bends  of  the  road  between 
stations  can  be  sketched  with  all  needful  accm-acy. 

During  the  progress  of  the  work  all  points  off  the  line  which  are  capa- 
ble of  being  located  by  intersection  must  be  located  by  sights  taken  from 
stations,  and  special  care  must  be  taken  to  connect  them  with  the  points 
located  by  the  secondary  triangulation,  in  order  to  afford  as  many  checks  as 
possible  to  the  accm-acy  of  the  traverse  line. 

Traverse  lines  should  close  with  but  ti-ifling  eiTor — an  eighth  of  an  inch 
upon  the  paper  in  a  distance  of  10  or  12  miles  is  as  great  an  eiTor  as  should 
be  permitted— and  all  errors  of  closm-e  should  be  shown.  No  line  should 
be  arbitrarily  closed  on  the  traverse  sheet. 

The  traverse  man  should  sketch  or  locate  all  country  houses,  should 
note  aU  road  intersections  and  all  raih'oad  crossings,  specifying  by  simple 


TRAVERSE  WORK.  89 

conventions  the  character  of  the  crossing,  whether  over,  under,  or  grade 
crossing.  He  should  similarly  describe  all  stream  crossings ,  distinguishing 
fords,  ferries,  and  bridges. 

MEASUREMENTS    OF    HEIGHT    IN    CONNECTION    WITH    TEAVERSE    LINES. 

Height  measm-ements  in  connection  with  traverse  lines  are  effected  in 
one  of  two  ways — either  by  vertical  angles  with  the  telescopic  alidade  or 
by  the  use  of  the  aneroid. 

In  regions  where  little  or  no  secondary  triangulation  can  be  done,  it 
becomes  necessary  to  accompany  certain  of  the  traverse  lines  by  profiles 
determined  by  vertical  angles.  Such  profiles  should  be  surveyed  at  inter- 
vals of  4  or  6  miles  where  the  contour  interval  is  20  feet,  and  at  intervals 
of  8  or  10  miles  where  it  is  50  feet. 

The  alidade  generally  used  in  running  these  profiles  is  of  a  small  com- 


FiG.  10 Small  Telescopic  Alidade. 

pact  form,  with  low  standards  and  short  ruler.  The  telescope  has  low 
power,  but  carries  a  good  vertical  arc  and  a  level.  The  arc  and  vernier 
are  graduated  to  single  feet  with  a  radius  of  a  mile,  instead  of  degi-ees  and 
minutes,  in  order  to  facilitate  computation.  This  graduation  is  made  on  the 
assumption  that  where  the  angle  is  less  than  5°  the  arc  and  the  tangent  do 
not  materially  differ. 

With  this  instrument  the  plan  of  the  traverse  is  run  precisely  as  above 
sketched,  except  that  a  rodman  is  frequently  employed.  In  running  the 
profile,  which  is  done  coincidently  with  the  plan,  the  points  sighted  for 
elevation  may  be  the  same  as  are  used  for  the  plan.  If  a  rodman  is  em- 
ployed, the  target  on  the  rod  should  be  set  at  the  height  of  the  instrument 
to  simplify  record  and  computation. 


90  A  MANUAL  OF  TOPOGEAPHIO  MJOTHODS. 

It  must  not  be  understood,  however,  that  it  is  at  all  necessary  that  the 
survey  of  the  profile  should  establish  the  height  of  all  the  points  located 
by  the  traverse.  The  profile  should  give  the  elevation  of  all  valleys  and 
summits,  and  of  all  road  crossings.  The  line  should  be  carried  forward 
and  these  points  ineasured  by  as  few  and  as  long  lines  of  sight  as  possible. 
Often  the  roof  of  a  house  will  furnish  a  datum  point  for  use  for  a  mile  or 
two.  Indeed,  in  an  open,  settled  country  the  line  can  frequently  be  carried 
forward  coutinuoiisly  by  using  housetops  as  targets. 

The  reduction  of  the  profile  must  keep  pace  with  the  field  work,  so 
that  on  arriving  at  a  check  point  the  amount  of  the  error  may  be  shown  at 
once.  If  this  is  not  more  than  one-fom'th  or  one-fifth  of  the  contour 
inten'^al,  it  is  not  considered  as  of  material  account.  If,  however,  it  reaches 
half  a  contour  interval,  the  work  should  be  examined,  and  if  the  error  be 
not  discovered  the  line  should  be  resurveyed. 

The  heights,  as  determined,  should  be  written  in  ink  upon  the  plane- 
table  sheet  in  their  proper  places. 

-      THE   ANEROID. 

In  the  great  majority  of  traverse  work  heights  are  measured  with 
aneroids.  The  aneroid  consists  of  a  vacuum  box  of  thin  coiTugated  metal, 
which  is  compressed  by  an  increase  and  expanded  by  a  decrease  in  the 
pressm-e  of  the  atmosphere.  A  ti'ain  of  mechanism  magnifies  this  trifling 
movement  enormously  and  moves  an  index  upon  a  graduated  dial.  This 
dial  is  graduated  to  feet  of  elevation  and  also  to  inches  of  barometric 
pressm-e. 

Several  sizes  of  aneroids  are  made;  that  having  a  diameter  of  2 J 
inches  is  on  the  whole  found  the  most  satisfactory. 

Owing  mainly  to  its  extreme  delicacy  the  aneroid  is  a  very  uncertain 
instiTiment.  It  should  be  used  difi'erentially  only,  and  for  small  diiferences 
in  height  and  small  intervals  of  time.  Its  indications  should  be  checked 
by  reference  to  known  elevations  whenever  opportunity  is  afforded  during 
the  day,  and  at  the  beginning  and  ending  of  each  day's  work. 

On  commencing  work  the  movable  scale  on  the  aneroid  should  be  set 
at  the  known  height  of  the  starting  point  and  a  note  made  of  its  reading 
on  the  inch  scale.     Elevations  should  then  be  read  du-ectly  from  the  scale 


THE  AFEEOID. 


91 


of  feet.  The  heights  of  all  points  along  the  line  of  traverse  which  will  be 
required  in  making  the  contour  sketch  should  be  read  and  written  upon  the 
traverse.  Every  depression  and  elevation,  road  crossing,  etc.,  should  thus 
be  measured.  There  is,  hoAvever,  no  necessity  for  reading  the  aneroid  at 
every  station  in  the  traverse.  It  will  merely  encumber  the  work  with  a 
mass  of  useless  data. 

Upon  reaching  a  check  point,  comparison  should  be  made  with  the  indi- 
cations of  the  aneroid.  If  the  difference  is  considerable — i  e.,  more  than  a 
contour  interval — the  error  should  be  distributed  backward  along  the  line  in 
proportion  to  distance.     If  it  is  small,  it  may  be  neglected. 


Fig.  U. —Aneroid. 


Fig.  12.— Works  of  the  Aneroid. 


In  all  this  work  notebooks  are  not  required,  except  as  a  convenient  form 
of  carrying  paper  upon  which  to  make  the  trifling  computations  required. 
The  plane-table  sheets  comprise  all  the  records  necessary.  The  work,  as 
it  progresses,  criticises  itself  by  its  closui-es  in  position  and  elevation,  and, 
wherever  necessary,  is  revised  immediately. 


ORGANIZATION  OF  PARTIES  AND  DISTRIBUTION  OF  WORK. 

Secondary  triangulation,  traversing,  measuring  of  heights,  and  sketch- 
ing are  commonly  carried  on  by-  one  party.  This  consists  of  the  chief  of 
party,  who  directs  all  the  operations,  and  who  does  all  the  sketching;  an 


92  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

assistant  who  carries  on  the  secondary  trianguhation,  selected  as  possessing 
special  fitness  for  that  work,  and  one,  two,  or  more  assistants  who  are 
engaged  in  traversing,  the  number  of  these  assistants  depending  upon  the 
rapidity  with  which  the  country  can  be  sketched  relative  to  the  rate  at  which 
the  traversing  progresses.  If  possible,  the  difierent  items  of  work  of  such  a 
partv  should  follow  one  another  in  a  certain  order.  The  secondary  trian- 
gulation  should  be  done  first  in  order  that  the  traverse  men  may  be  furnished 
with  positions  and  heights  for  locating  and  checking  their  traverse  lines.  The 
traversing  should  follow,  in  order  that  all  the  control  may  be  furnished  to 
the  chief  of  party  for  his  use  in  sketching.  This  order,  which  is  followed  as 
closely  as  practicable,  requires  that  the  members  of  the  party  be  scattered 
over  a  considerable  area  of  coimtry,  and  if  they  are  living  in  camp  it 
requires  that  they  remain  away  from  it  a  considerable  part  of  the  time,  or 
else  that  a  large  amount  of  traveling  be  done  in  order  to  reach  camp  at  night. 
Wliere  they  are  not  living  in  camp,  the  most  economical  disposition  is  to 
scatter  them  at  various  places  within  their  fields  of  work.  In  any  case,  con- 
stant communication  must  be  had  between  the  chief  of  party  and  his  assist- 
ants, in  order  that  they  may  work  in  accord. 

STADIA  MEASUREMENT. 

Under  certain  circumstances  it  is  found  advisable  to  use  the  stadia 
method  for  measuring  distances  in  place  of  the  wheel.  This  is  the  case 
where  lines  are  to  be  run  without  reference  to  roads,  and  consequently 
where  the  wheel  cannot  be  employed  with  advantage.  It  has  been  used,  too, 
in  southeiTL  Louisiana,  where  peculiar  methods  of  work  imposed  by  the  nature 
of  the  topography  have  made  its  employment  economic.  The  instniment  used 
for  the  stadia  or  telemeter  method  of  measuring  distances  may  be  anything 
cari'jnng  a  telescope.  To  the  reticule  of  the  telescope  are  added  two  or  more 
fixed  horizontal  wires  placed  at  a  certain  distance  apart.  A  rod  or  board 
subdivided  to  suit  the  interval  between  the  wires  and  painted  in  glaring 
colors  forms  part  of  the  outfit.  When  this  rod  is  set  up  at  a  distance  from 
the  telescope,  that  distance  is  ascertained  from  the  number  of  subdivisions 
of  the  rod  which  are  included  between  the  wires  of  the  telescope,  the  value 
of  each  division  of  the  rod  being  known.     Upon  the  Geological  Survey  cer- 


STADIA  MEASUEEMBNTS.  93 

tain  theodolites  and  telescopic  alidades  are  equipped  with  stadia  wires. 
These  wires  are  three  in  number,  the  intervals  between  them  being  equal. 
The  rods  are  14  feet  in  length  and  hinged  so  as  to  close  to  7  feet.  The 
intervals  upon  the  rods  are  of  one  foot  each.  The  wires  in  the  telescope  are 
so  spaced  that  when  the  rod  is  at  a  distance  of  100  feet,  the  space  between 
the  two  extreme  wii-es  will  subtend  one  foot  on  the  rod.  At  a  distance  of 
1,400  feet,  therefore,  this  space  will  subtend  the  entire  length  of  the  rod, 
while  at  a  distance  of  2,800  feet  two  adjacent  wires  in  the  telescope  will 
subtend  the  entire  length  of  the  rod.  Distances  less  than  100  feet  are  esti- 
mated by  means  of  the  fractional  2Dart  of  a  foot  upon  the  rod,  which  is 
included  between  the  wii'es.  The  distances  are  read  off  upon  the  rod  by 
the  surveyor  at  the  instrument. 

In  measuring  distance  upon  slopes,  correction  must  be  made  to  reduce 
the  inclination  measured  to  horizontal  distance.  Tables  for  this  reduction 
are  to  be  found  in  Bulletin.  Where  the  slope-  is  slight  it  is  not  regarded  as 
necessary  to  make  this  reduction,  especially  where  there  are  frequent  points 
for  checking  and  correcting  the  line. 

The  rod  may  be  used  also  for  measurement  of  the  profile  of  a  line. 
For  this  purpose,  a  point  should  be  marked  upon  it  at  the  same  height  as 
the  telescope  of  the  instrument  and  vertical  angles  taken  to  this  point. 

The  work  which  has  been  can-ied  on  in  southern  Louisiana  is  peculiar 
in  the  fact- that  the  slopes  are  extremely  gentle,  requiring,  in  order  to  show 
the  relief  at  all,  a  contour  interval  not  greater  than  5  feet.  For  the  location 
of  contours  of  so  small  an  interval,  even  -vertical  angles  are  not  sufficiently 
accurate,  and  the  work  of  measurement  is  effected  by  spirit  level.  The 
instrument  used  is  a  theodolite  of  compact  and  simple  form,  to  which  the 
name  of  gradienter  has  been  applied,  which  is  equipped  with  stadia  wires. 
The  low  ridges  which  accompany  the  streams  of  this  region  and  which  form 
all  the  relief  are  located  by  means  of  lines  run  approximately  at  right  angles 
to  the  streams  from  their  banks  down  to  the  swamps  on  either  side.  Dis- 
tances are  obtained  by  stadia  and  differences  of  elevation  by  using  the 
gradienter  as  'a  wye  level,  and  the  stadia  rod  as  a  level-rod. 


94  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

THE    CISTERN    BAROMETER. 

lu  work  bavirig  a  large  contour  iuterval,  50  feet  or  more,  the  cistern 
barometer  is  used  to  some  extent,  though  not  as  much  as  formerly.  Its  use 
is  now  confined  to  the  work  in  the  far  West,  where  it  is  employed  in  the 
determinations  of  heights  of  points  in  the  valleys  not  easily  reached  by 
vertical  angles. 

The  barometer  is  an  instrument  for  measuring  the  pressui-e  of  the 
atmosphere.  At  the  level  of  the  sea  this  pressure  of  about  15  pounds  per 
square  inch  supports  a  column  of  mercury  about  30  inches  in  height.  As 
one  rises  above  sea  level  and  leaves  a  portion  of  the  atmosphere  behind 
him  the  pressure  diminishes  and  the  column  of  mercury  sustained  by  it  is 
of  less  height. 

The  cistern  barometer,  in  its  most  portable  form,  is  made  by  H.  J. 
Green.  It  consists  of  a  cistern  into  which  dips  the  lower  open  end  of  a 
glass  tube  31  or  32  inches  in  length,  the  whole  being  inclosed  in  a  brass 
case.  The  cistern  consists  of  a  number  of  parts,  including  a  short  glass 
cylinder,  below  which  is  fitted  the  inverted  frustum  of  a  hollow  cone  of 
boxwood.  This  is  succeeded  by  a  second  frustum,  placed  upright,  from 
the  lower  end  of  which  depends  a  bag  of  buckskin.  The  bottom  of  the 
latter  is  raised  or  lowered  by  means  of  a  screw  in  the  brass  case  of  the 
cistern.  The  cisteiTi  is  closed  at  the  top  by  a  boxwood  ring,  which  is  fitted 
to  the  top  of  the  glass  cylinder.  By  means  of  an  annular  piece  of  kid, 
which  is  securely  lashed  to  the  boxwood  ring  and  to  the  barometer  tube, 
the  cistern  and  the  tube  are  connected.  From  the  under  sm-face  of  the 
boxwood  ring  depends  an  ivory  point  about  a  quarter  of  an  inch  in  length. 
Upon  the  brass  casing  of  the  tube  is  a  graduation  into  inches  and  twentieths, 
by  which,  with  the  aid  of  A^erniers,  the  scale  may  be  read  to  0.002  of  an 
inch.  To  this  brass  case  is  attached  a  thei-mometer,  for  indicating  the  tem- 
peratm-e  of  the  instrument.  For  carriage  the  barometer  is  placed  in  a 
wooden-  case  fitted  to  its  shape,  and  this  in  turn  in  a  case  of  heavy  leather 
fitted  with  a  shoulder  strap.  It  should  always  be  carried  in  an  inverted 
position. 

To  read  the  instrument  it  should  be  hung  where  it  can  swing  freely. 
Then,  by  lowering  the  screw  at  the  bottom,  di-op  the  mercury  in  the  cistern 


THE  CISTEEN  BAEOMETEE.  95 

until  its  top  just  touches  the  ivory  point  above  mentioned.  This  can  be 
best  effected  by  making  the  ivory  point  and  its  reflection  from  the  surface 
of  the  mercury  barely  touch  one  another.  Then  move  the  vernier  until  its 
bottom  is  just  tangent  to  the  convex  top  of  the  mercury  in  the  tube. 

The  vernier  is  read  like  other  verniers  and  requires  no  special  expla- 
nation. Besides  reading  the  height  of  the  column  of  mercury  in  the 
barometer,  it  is  necessary  to  read  its  temperature  by  means  of  the  attached 
thermometer,  and  also  the  temperature  of  the  air  by  means  of  a  thermom- 
eter hung  in  the  shade. 

The  barometer  is  used  differentially — that  is,  the  difference  in  height 
between  two  points  is  determined  by  the  difference  in  the  indications  of  two 
barometers,  one  at  each  point.  In  order  to  obtain  the  height  above  sea  level 
of  one  of  these  points,  that  of  the  other  must  be  known.  The  latter  is  called 
the  base  station,  and  its  altitude  should  be  determined  either  by  leveling  or 
by  a  long  series  of  barometric  observations  referred  to  some  other  point 
whose  altitude  has  been  established.  The  proper  selection  of  a  base  station 
or  a  system,  of  base  stations  for  reference  of  work  to  be  done  in  a  certain 
locality  is  a  matter  involving  considerable  judgment  and  a  knowledge  of 
the  peculiar  errors  to  which  the  barometer  is  liable,  as  well  as  a  knowledge 
of  the  topography  of  the  country  and  its  probable  influence  upon  the 
fluctuations  of  barometric  pressure.  The  base  station  should  be  near  the 
middle  of  the  area.  If  but  one  base  station  is  employed,  it  should  be  near 
the  middle  altitude  of  the  region.  If  two  be  used,  one  should  be  at  the 
altitude  of  the  low  or  valley  country  and  the  other  should  in  altitude  be 
near  the  high  summits.  In  the  Hayden  survey  of  Colorado  tln-ee  base 
stations  were  employed  at  once — one  at  Denver,  at  an  altitude  of  5,300 
feet;  one  at  Fairplay,  10,000  feet,  and  one  near  the  summit  of  Mount 
Lincoln,  14,200  feet.  To  these  base  stations  were  referred  severally  those 
observations  taken  at  points  most  nearly  approaching  them  in  height. 

Comparisons  should  be  made  between  the  readings  of  ihe  base  barome- 
ter and  the  readings  of  those  to  be  used  in  the  field.  These  comparisons 
should  be  made  with  the  barometers  hung  side  by  side  and  should  be  made 
in  full — i.  e.,  by  lowering  the  mercury  from  the  tubes,  its  level  in  the  cistern 
to  the  ivory  point,   and  resetting  the  verniers  at  each  reading — and  the 


96  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

attached  thermometei's  slioiild  be  read.  Both  barometers  should  be  read  by 
the  same  observer.  A  half  dozen  observations  made  at  intervals  of  half  an 
hour  will  answer  as  well  as  a  greater  number.  Such  comparisons  should, 
if  practicable,  be  made  at  the  beginning  and  the  end  of  the  season,  wlien- 
ever  a  new  tube  is  put  into  either  barometer,  or  after  any  repairs  to  either 
instrument. 

The  discrepancies  between  the  readings  of  two  barometers  are  due  to 
several  causes,  among  which  are  differences  in  setting  of  the  scale  of  inches, 
differences  in  the  caliber  of  the  tubes,  causing  difiPerent  amounts  of  capillar- 
ity^ and  differences  in  the  perfection  of  the  vacuums  in  the  tubes.  Differ- 
ences due  to  the  first  two  are  generally  trifling,  amounting  to  but  a  few 
thousandths  of  an  inch.  If  large  discrepancies  exist,  they  are  usually  due 
to  the  last  cause,  and  this  should  be  corrected. 

The  cistern  barometer  is  a  very  frail  instrument,  and  although  in  the 
mountain  form  it  is  protected  from  accident  as  thoroughly  as  possible,  still 
tubes  are  not  infrequently  broken  while  in  the  field.  It  is  necessary,  there- 
fore, to  provide  the  requisite  means  for  making  repairs,  such  as  sealed  tubes, 
distilled  mercury,  etc.  When  a  tube  is  broken,  the  barometer  should  be 
opened  at  once,  and  the  mercury  poured  out,  in  order  to  prevent  it  from 
dissolving  the  screws  and  other  brass  work  of  the  instrument. 

The  work  of  filling  and  replacing  a  tube  is  a  delicate  operation.  After 
taking  the  barometer  to  pieces,  the  new  tube  should  be  opened  by  breaking 
off  the  small  end,  the  break  being  made  at  a  distance  from  the  strictm-e  equal 
to  that  upon  the  old  tube.  It  should  be  effected  by  cutting  it  around  with 
a  sharp  file,  when  a  little  pressure  will  cause  it  to  break;  then  the  edge  of 
the  break  should  be  smoothed  with  a  file.  The  collar  which  forms  the  top 
of  the  cistern  should  then  be  lashed  on  to  the  tube  at  the  strictm-e.  The 
mercury  to  be  used  should  be  very  pure,  and  to  clear  it  from  mechanical 
impurities,  it  should  be  strained  tln-ough  chamois  skin  immediately  before 
use.  It  should  then  be  poured  into  the  tube  through  a  paper  funnel,  and 
the  tube  filled  to  within  an  inch  of  the  top.  Then,  covering  the  open  end 
of  the  tube  with  the  finger,  protected  by  a  piece  of  kid,  invert  the  tube, 
letting  the  bubble  of  air  slowly  traverse  the  tube  up  and  down  for  the  pur- 
pose of  collecting  the  minute  air  bubbles  which  may  have  remained  in  the 


THE  CISTERN  BAEOMETER.  97 

tube.  Do  this  repeatedly,  if  necessary,  until  the  mercury  appears  perfectly 
clear  of  bubbles.  Then  fill  the  tube  with  merciu-y,  ch'awing  out  with  a  straw 
any  bubbles  that  may  then  be  near  the  top.  Invert  the  tube  in  the  case,  put 
on  the  glass  ring  and  the  upper  cone  of  the  cistern,  and  screw  them  together. 
Then  fill  the  cistern  with  mercury,  put  on  the  lower  cone,  with  the  bag  and 
the  brass  cover,  and  the  work  is  complete.  The  test  of  a  satisfactory  result 
is  the  sound  made  by  the  column  of  mercury  as  it  strikes  the  top  of  the  tube. 
If  there  is  a  sharp  metallic  click  the  vacuum  is  good,  but  if  the  sound  is 
muffled  the  vacuum  must  be  improved.  It  is  well  to  warm  the  mercury 
before  pouring  it  into  the  barometer,  in  order  to  drive  out  any  moisture  in 
it.     This  is  especially  ad-sdsable  if  the  atmosphere  is  damp  at  the  time. 

It  is  by  some  thought  advisable  to  boil  the  mercury  in  the  tube  during 
the  operation  of  filling.  This  is  usually  done  over  an  alcohol  lamp,  two  or 
thi-ee  inches  of  mercury  being  poured  into  the  tube  at  a  time  and  brought 
to  a  boil  until  the  tube  is  filled.  The  mercury  which  is  to  be  poured  into 
the  cistern  is  then  also  boiled.  This  is  a  very  delicate  and  tedious  operation, 
and  is  attended  with  much  risk  to  the  tubes.  Its  utility  is  questionable, 
inasmuch  as  the  mercury  in  the  barometer  is  exposed  to  the  atmosphere  and 
soon  contains  as  much  moisture  as  before. 

It  often  becomes  necessary  to  clean  the  sui-face  of  the  mercury  in  the 
cistern.  To  do  this,  take  off  the  lower  cone  of  the  cistern ;  then,  placing 
the  finger,  protected  by  a  piece  of  kid,  over  the  open  end  of  the  tube, 
invert  the  barometer  slowly  and  pour  out  the  mercury  from  the  cistern. 
Strain  it  tln'ough  chamois  skin,  replace  it  in  the  cistern,  and  put  the  latter 
together  again. 

Observations  at  the  base  stations  should,  whenever  practicable,  be 
made  hourly  from  7  a.  m.  to  9  p.  m.,  in  order  to  insure  having  base  obser- 
vations coincident  with  those  taken  in  the  field.  When  not  practicable  to 
do  this,  they  should  be  made  at  7  a.  m.,  2,  6,  and  9  p.  m.  Each  observation 
should  include  the  reading  of  the  attached  and  detached  thermometers. 
Whenever  the  observations  at  a  station  of  the  U._  S.  Weather  Bureau  are 
available,  they  may  be  used  as  base  records.  In  most  cases,  howevei,  these 
observations  are  made  with  barometers  reading  only  to  one-hundi-edth  of 
an  inch,  but,  upon  proper  application,  the  Weather  Bureau  has  in  all  cases 

MON  XXII 7 


98  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

substitiited  barometers  reading  more  minutely  in  order  to  meet  the  require- 
ments of  the  work  of  this  Sm-vey. 

In  field  work,  barometers  should  be  read  at  each  camp  hourly  during 
the  daytime,  if  practicable,  or,  if  not,  at  such  hours  as  to  correspond  with 
the  readings  at  the  base  station  and  with  readings  made  by  the  topographer 
in  the  course  of  his  work,  having  in  view  the  use  of  the  camp  as  a  sort  of 
secondary  base  station.  The  topographer  or  his  assistant  should  read  the 
barometer  on  all  stations,  and  at  all  important  points  the  heights  of  which 
cannot  be  more  easily  obtained  by  vertical  angles. 

Measurements  of  height  made  with  cistern  barometers  are  subject  to 
periodic  and  accidental  errors.  The  periodic  errors  are  probably  due  to 
imperfections  in  the  formulas  and  constants  used  in  the  reduction.  Many 
attempts  both  from  theoretical  and  practical  points  of  view  have  been  made 
to  remedy  these  defects,  but  thus  far  without  success.  The  accidental  errors 
are  due  to  eiTors  of  obseiwation  and  to  local  differences  in  the  pressure  of 
the  an-  at  the  points  at  which  observations  are  made.  Where  the  hori- 
zontal distance  between  the  two  stations  compared  is  great,  such  differences 
may  be  correspondingly  great,  and  the  same  is  true  where  there  is  a  con- 
siderable difference  of  elevation  between  the  two  stations. 

Under  favorable  circumstances  barometric  observations  should  give  the 
height  within  a  score  of  feet.  Where  the  circumstances  are  unfavorable — as, 
for  instance,  where  there  is  a  great  difference  of  elevation  between  the  two 
stations  or  a  great  horizontal  distance  between  them — the  error  may  be  large, 
reaching  100  feet,  and  even  in  extreme  cases  200  feet. 

REDUCTION   OF    BAROMETRIC    OBSERVATIONS. 

The  pressure  of  the  atmosphere  at  the  sea  level  is  approximately  15 
pounds  per  square  inch,  or  is  equivalent  to  that  of  a  column  of  mercury 
30  inches  in  height.  With  elevation  the  pressin-e  diminishes,  but  not  in  a 
simple  ratio  to  the  altitude,  as  would  be  the  case  if  all  the  strata  had  the 
same  density.  The  density  is  proportional  to  the  pressure,  and  as  the 
pressure  upon  each  layer  is  produced  by  the  body  of  air  above  it,  it  follows 
that  each  succeeding  layer  of  air  is  less  dense  than  that  which  underlies 


THE  CISTERiT  BAROMETER.  99 

it.     The  relation  between  altitude  and  atmospheric  pressure,  as  stated  by 
Gilbert,  is  as  follows: 

The  difl'ereuce  in  height  of  any  two  localities  is  equal  to  a  certain  constant 
distance  multiplied  by  the  difference  between  the  logarithms  of  the  air  pressures  at 
the  two  localities. 

This  relation  gives  the  first  and  principal  term  in  the  various  tables  for 
the  reduction  of  barometric  work.  Different  determinations  of  the  constant 
distance,  known  as  the  "pressure  constant,"  have  been  made,  and  these 
different  pressure  constants  cause  the  principal  differences  in  the  various 
tables  in  use. 

Of  the  different  sets  of  tables  yielding  good  results,  the  most  con- 
venient for  use  are  those  known  as  Guyot's.  They  are  published  in  the 
Smithsonian  Miscellaneous  Collections,  No.  13,  and  republished  in  this 
volume  tables  I  to  V.  These  tables  are  derived  from  the  formula  of  La 
Place  and  use  his  coefficients.  The  formula,  reduced  to  English  measures, 
is  as  follows : 


Z  =log.  A  X  60158.6  English  feet  < 


^  ^       900 

(1+  0.0026  cos  2  L) 

,       Z  +  52252  h        ) 

'  +20886860+10443430  ) 


h   rr  the  observed  height  of  the  barometer  ■\ 

r   —  the  temperature  of  the  barometer  >  at  the  lower  station; 

t    ^  the  temperature  of  the  air  } 

h'  z^  the  observed  height  of  the  barometer  \ 

r'  zz  the  temperature  of  the  barometer  >  at  the  upper  station. 

t'   izi  the  temperature  of  the  air  ) 

Z  —  the  difference  of  level  between  the  two  barometers ; 

L  zz  the  mean  latitude  between  the  two  stations; 

H  =:  the  height  of  the  barometer  at  the  upper  station  reduced  to  the 

temperature  of  the  barometer  at  the  lower  station ;  or, 
n  =  h'  {1  +  0.00008967  (r  —  r')}. 

Table  I  gives,  in  English  feet,  the  value  of  log.  H  or  h  X  60158.6  for 
every  hundredth  of  an  inch,  from  12  to  31  inches  in  the  barometer,  together 


100         'A  MANUAL  OF  TOPOGKAPHIC  METHODS. 

witli  the  value  of  the  additional  thousandths,  in  a  separate  column.  These 
values  have  been  diminished  by  a  constant,  which  does  not  alter  the  differ- 
ence required. 

Table  II  gives  the  correction  2.343  feet  X  C''  —  ^')  ^i"  the  difference  of 
the  temperature  of  the  barometers  at  the  two  stations,  or  r  —  t'.  As  the 
temperature  at  the  upper  station  is  generally  lower,  r  —  r'  is  usually  posi- 
tive and  the  correction  negative.  It  becomes  positive  Avheu  the  temperature 
of  the  upper  barometer  is  higher  and  t  —  t'  negative.  When  the  heights 
of  the  barometers  have  been  reduced  to  the  same  temperatures,  or  to  the 
freezing  point,  this  table  will  not  be  used. 

Table  IV  shows  the  correction  D'  2088686O  *^  ^^  fipplied  to  the 
approximate  altitude  for  the  decrease  of  gravity  on  a  vertical  acting  on 
the  density  of  the  mercurial  column.     It  is  always  additive. 

h 
Table  V  furnishes  the  small  con-ection  ^ -,,„.--  for  the  decrease  of 

lU4:4o4:OU 

gravity  on  a  vertical  acting  on  the  density  of  the  air ;  the  height  of  the 
barometer  h  at  the  lower  station  representing  its  approximate  altitude. 
Like  the  preceding  correction,  it  is  always  additive. 

USE    OF    THE    TABLES. 

In  Table  I  find  first  the  numbers  corresponding  to  the  observed  heights 
of  the  barometer  h  and  h'.  Suppose,  for  instance,  h  zn  29.345  in. ;  find  in  the 
first  column  on  the  left  the  number  29.3;  on  the  same  horizontal  line,  in  the 
column  headed  .04,  is  given  the  number  corresponding  to  29.34  z:  28121.7; 
in  the-' last  column  but  one  on  the  right,  we  find  for  .005  =  4.5,  or  for 
29.345  =  28126.2.     Take  Ukewise  the  value  of  h',  and  find  the  difference. 

If  the  barometrical  heights  have  not  been  previously  reduced  to  the 
same  temperature  or  to  the  freezing  point,  apply  to  the  difference  the  cor- 
rection found  in  Table  II  opposite  the  number  representing  r  —  r';  we  thus 
obtain  the  approximate  difference  of  level,  D. 

For  computing  the  correction  due  to  the  expansion  of  the  air  according 

to  its  temperature,  or  D  X  (  q^T )  make  the  sum  of  the  tempera- 
tures, subtract  from  that  sum  64;  multiply  the  rest  into  the  approximate 


PUBLIC  LAND  SUEVEYS.  101 

difference  D  and  divide  the  product  by  900.  This  coiTSction  is  of  the  same 
sign  as  (t  +  f  —  64).  By  applying  it,  we  obtain  a  second  approximate  dif- 
ference of  level,  D'. 

In  Table  III,  with  D'  and  the  mean  latitude  of  the  stations,  find  the 
correction  for  variation  of  gravity  in  latitude,  and  add  it  to  D',  paying  due 
attention  to  the  sign. 

In  Table  IV  with  D',  and  in  Table  V  with  D'  and  the  height  of  the 
barometer  at  the  lower  station,  take  the  con-ections  for  the  decrease  of 
gravity  on  a  vertical,  and  add  them  to  the  approximate  difference  of  level. 

The  sum  thus  found  is  the  true  difference  of  level  between  the  two 
stations,  or  Z;  by  adding  the  elevation  of  the  lower  station  above  the  level 
of  the  sea,  when  known,  we  obtain  the  altitude  of  the  upper  station. 

UTILIZATION  OF  THE  WORK  OF  THE  PUBLIC  LAND  SURVEYS. 

In  all  the  states  and  territories  except  the  original  thirteen,  together  with 
Vermont,  Kentucky,  Tennessee,  Texas,  and  Alaska,  the  public-land  sur- 
veys have  been  carried  on,  and  many  of  these  states  have  been  entii-ely 
covered  by  these  surveys. 

These  surveys  were  made  for  the  purpose  of  dividing  the  land  into 
parcels  suitable  for  sale  or  other  disposition,  and  with  httle  reference  to 
map  purposes.  The  work  differs  widely  in  quality  in  different  parts  of  the 
country,  in  some  regions  being  very  bad,  in  others  of  high  quality.  6rener- 
ally  speaking,  the  later  work  is  much  the  bettei*. 

This  work  is  extensively  used  by  the  Geological  Survey  as  an  aid  in 
the  preparation  of  its  maps.  The  extent  to  which  it  is  utiHzed,  and  the 
methods  employed  in  using  it,  will  be  detailed  in  this  chapter.  Before 
proceeding  with  this,  however,  it  is  desirable  to  describe  the  methods 
by  which  this  work  has  been  and  is  carried  on. 

The  system  of  subdivision  is  an  extremely  simple  one.  It  consists,  first, 
in  the  division  of  the  land  into  large  blocks,  the  division  of  these  blocks  into 
townships,  approximately  6  miles  on  a  side,  and  the  subdivision  of  these 
townships  into  sections,  each  containing  about  1  square  mile.  Fm-ther 
subdivision  of  these  sections  into  quarter  sections,  or  even  smaller  areas,  has 
been  done  by  private  surveyors. 


102  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

The. supervision  of  the  surveys  is  vested  in  surveyors-general,  one  in 
each  state  or  territory  in  which  such  surveys  are  being  carried  on.  The 
surveys  are  made  by  contract,  at  certain  stated  prices  per  linear  mile,  and 
are  subject  to  examination  by  salaried  officers  of  the  Land  Office. 

The  initial  work  consists  in  the  measurement  of  a  principal  meridian 
and  a  base  line,  their  intersection  being  the  initial  point  of  the  survey. 
These  lines  are  run  with  considerable  care.  The  principal  meridian  may 
be  run  both  northward  and  southward  from  the  initial  point,  and  the 
instructions  require  that  observations  be  made  for  azimuth  at  intervals  not 
greater  than  12  miles,  and  that  the  line  be  double  chained,  two  sets  of  chain- 
men  being  employed  for  that  purpose.  In  measuring  a  base  line,  which  is 
to  follow  as  closely  as  possible  a  parallel  of  latitude,  in  case  the  theodolite 
be  used-it  is  to  be  run  by  means  of  a  succession  of  tangents  to  the  parallel, 
not  exceeding  12  miles  in  length.  At  intervals  of  half  a  mile  a  point  on  the 
parallel  is  marked  by  offsets  from  the  tangent  line,  and  at  the  end  of  12 
miles  a  new  tangent  is  commenced.  In  case  it  be  run  by  solar  compass,  it 
must  be  checked  by  latitude  observations  at  intervals  of  12  miles.  The  base 
line  may  be  run  either  east  or  west  from  the  principal  meridian.  At  inter- 
vals of  24  miles  on  the  base  line  auxiliary  meridians  are  run  in  the  same 
manner  as  prescribed  for  the  principal  meridian,  and,  at  intervals  of 
24  miles  on  the  meridian,  correction  lines  are  run  east  and  west  in  a 
similar  manner.  It  is  only  recently  that  the  interval  between  guide  merid- 
ians and  coiTection  lines  has  been  reduced  to  24  miles,  or  4  townships. 
Heretofore  the  intervals  have  differed  at  different  times,  but  have  in  all  cases 
been  greater.  These  lines  are  run  with  a  solar  compass  or  theodolite,  and 
never  in  later  years  with  the  ordinary  compass,  and  all  these  lines  double 
chained. 

By  this  means  the  country  is  divided  into  approximate  squares  24  miles 
on  a  side.  Each  such  area  is  then  divided  into  townships  approximately  6 
miles  on  a  side.  The  east  and  west  sides  of  these  townships  are  meridians 
which  are  run  northward  from  the  base  line  or  from  the  correction  line, 
ha^ang  a  breadth  upon  the  base  or  correction  line  of  6  miles,  but  decreasing 
in  breadth  with  the  convergence  of  the  meridians.  The  north  and  south 
sides  of  the  townships  may  be  run  east  or  west,  as  the  case  may  be.     The 


PUBLIC  LAND  SYSTEM.  103 

east  and  west  township  lines  as  at  first  run  are  simple  random  lines,  wHch 
are  corrected  backward  in   order  to   suit   the  positions  of  the  township 
corners,  as  determined  upon  the  guide  meridians  and  north  and  south  town- 
ship lines      The  township  lines  are  all  run  with  a  solar  compass  or  transit, 
and  double  chaining  is  not  required.    The  east  and  west  sides  of  the  sec- 
tions are  run  in  all  cases  northward,  while  the  north  and  south  sides  may  be 
run  either  east  or  west.     As  in  running  township  lines,  the  first  east  and  west 
and  north  and  south  lines  in  the  northern  tier  of  sections  are  merely  random 
lines    to  be  corrected  backward,  the  mile  posts  upon  the  township  lines 
beino-  reo-arded  as  the  final  locations  of  the  section  comers.     In  running  the 
sectionlines  the  quarter-section  corners  are  marked,  but  the  lines  are  not  run 
by  the  Government  surveyors.     The  accumulated  errors  in  the  subdivision 
of  the  township  are  thrown  into  the  northern  and  western  tiers  of  sections. 
Surveys  have  been  started  from  numerous  initial  points,  involving  the 
measurement  of  a  number  of  principal  meridians  and  base  lines.     No  system 
has  been  followed  in  the  an-angement  of  principal  meridians  and  base  lines, 
or  in  the  subdivision  of  the  country  with  respect  to  them. 

In  making   these  surveys,  topography  is  mapped   to   but  a  limited 
extent      The  positions  of  all  streams  are  obtained  at  the  points  of  crossing 
of  the  hnes-i.  e.,  at  intervals  of  a  mile.     The  same  is  the  case  with  roads. 
All  streams  of  importance,  however,  are  traversed,  and,  in  the  case  of  navi- 
gable streams,  both  banks  are  traversed  separately.     The  margins  of  all 
lakes  and  ponds  of  magnitude  are  traversed,  and  the  outlines  of  all  swampy 
and  marshy  areas  are  indicated.     Indeed,  were  the  work  done  thoroughly 
everywhere,  there  would  be  obtained  material  for  a  map  fairly  accurate  m 
details  of  the  horizontal  elements.     Practically,  however,  the  degree  of  ful- 
ness varies  with  the  surveyor.     In  many  cases  the  plats  are  sufficiently 
full  of  detail  for  maps  upon  a  scale  of  2  miles  to  an  inch,  and  m  some 
cases  for  a  scale  even  larger.     In  other  cases,  over  considerable  areas,  the 
drainage  represented  is  exceedingly  scanty.     In  some  townships  few  or  no 
streams  are  represented.     In  other  words,  for  mapping  purposes,  the  work 
is  by  no  means  uniform  in  quality.     Furthermore,  no  attempt  has  hereto- 
fore been  made  to  obtain  correct  positions.     Most  of  the  initial  points  of  the 
survey  were  assumed  arbitrarilv,  and  their  positions  in  latitude  and  longi- 


104  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

tude  have  never  been  determined.  Another  and,  for  mapping  purposes, 
important  element  which  is  wanting  in  this  work  is  the  relief.  In  some 
cases  aneroid  observations  have  been  taken  along  the  lines  of  survey,  but 
they  were  never  used  for  the  purpose  of  drawing  contours. 

The  plats  are  prepared  in  duplicate,  one  copy  being  retained  at  the 
local  land  office  and  the  other  deposited  in  the  central  office  at  Washing- 
ton. They  are  now  being  photolithographed,  and  a  limited  number 
printed  of  each.  These  plats  are  upon  a  scale  of  2  inches  to  a  mile 
They  show  the  subdivisions  of  the  townships  with  their  areas.  They  show, 
also  the  streams,  roads,  swamps,  lakes,  timber,  and  prairie  as  they  existed 
at  the  time  of  survey.  Relief  is  but  feebly  expressed.  If  any  attention  is 
paid  to  it,  it  is  indicated  by  crude  hachures. 

This  work  is  of  service  mainly,  if  not  entirely,  in  furnishing  secondary 
locations.  Its  value  for  this  purpose,  however,  differs  widely.  In  some 
regions  it  is  not  sufficiently  trustworthy  to  be  used,  even  when  closely 
controlled  b}-  triangulation.  In  forest-covered  or  broken  country  it  is  often 
difficult  to  find  the  corners,  so  that  it  becomes  necessary  to  supplement  the 
few  discovered  by  traverses  connecting  one  with  another.  This  has  been 
the  case  with  the  sm-veys  in  Missouri.  In  open  countiy,  on  the  other  hand, 
where  the  surveys  are  of  good  quality,  they  furnish  a  complete  and  admi- 
rable system  of  minor  location,  often  obviating  entkely  the  necessity  of 
making  any  horizontal  locations,  aside  from  the  primary  work  necessary 
to  eliminate  the  accumulated  errors  of  the  system.  In  Iowa,  Illinois,  and 
Wisconsin,  traversing  is  done  only  to  a  limited  extent  and  for  the  purpose 
of  locating  the  details  of  what  are  called  "diagonal"  roads — that  is,  roads 
not  upon  section  lines.  The  common  practice  of  constructing  roads  upon 
section  lines,  which,  in  the  prairie  states,  has  grown  out  of  this  plan  of  sub- 
division, aids  greatly  in  the  work  of  survey.  This  system  of  roads  is  highly 
developed  in  Kansas,  where,  by  state  law,  every  section  line  may  have  a 
road  upon  it.  This  fact,  coupled  with  the  rectangiilar  subdivision  of  the 
sections  into  quai-ters,  80's,  and  40's,  marked  by  fences  or  hedges,  and  the 
fact  that  all  these  subdivisions  are  indicated  upon  county  maps,  renders  the 
work  in  this  state  a  simple  ma-tter,  while  the  resulting  map  is  admirably 
controlled.     The  same  is  true  of  Nebraska  and  the  Dakotas,  as  far  as  settle- 


PUBLIC  LAND  SUEVEYS.  105 

■  ments  have  extended  westward,  while  Wisconsin,  Illinois,  and  Iowa  present 
conditions  almost  as  favorable. 

The  piiblic-land  surveys  are  corrected  either  by  extending  over 
them  belts  of  triangulation  or  by  primary  traverses.  When  the  former  is 
employed,  it  is  unnecessary  to  cover  the  area  with  triang-ulation.  It  is 
sufficient  to  restrict  it  to  belts  of  simple  figures,  such  as  triangles  or  quadii- 
laterals,  such  belts  being  75  to  100  miles  apart. 

Each  triangulation  station  should  be  connected  by  the  simplest  and 
most  direct  method  with  the  nearest  section  corner  of  the  land  surveys.  This 
is  done  generally  by  measuring  the  direction  and  chaining  the  distance, 
although  it  may  be  necessary  to  run  a  short  traverse,  or  even  a  bit  of  minor 
triangulation,  in  order  to  reach  the  section  corner.  In  this  way  connection 
is  made  with  the  land  surveys  at  intervals  of -10  or  16  miles  along  the  belt  of 
the  triangulation.  These  locations  are  of  course  supplemented  by  any  other 
accurate  locations  which  may  have  been  made  in  the  region  under  survey. 

When  primary  traverses  are  employed  for  control,  connection  should 
be  made  with  all  section  and  township  lines  crossed,  the  distance  along  the 
line  to  the  nearest  corner  should  be  measured,  and  the  direction  of  the  line 
relative  to  the  courses  of  the  traverse  should  be  measru-ed. 

In  open  country,  where  the  public-land  surveys  are  of  good  quality, 
as  above  desciibed,  the  work  of  the  topographic  parties  is  reduced  to  the 
measurement  of  heights,  and  sketching.  All  the  roads  are  matters  of  public 
record  and  are  obtained  from  the  county  officers.  The  same  is  true  of  the 
plats  of  all  towns  and  the  plans  and  profiles  of  all  raih-oads.  These  are 
obtained  and  placed  upon  outline  plats  of  the  townships,  upon  a  scale  double 
that  of  which  the  maps  are  to  be  published. 

Heights  are  measured  with  the  vertical  cu'cle  and  by  aneroid,  except  in 
Illinois,  where,  the  contour  interval  being  10  feet,  the  vertical  circle  only  is 
used. 

Where  both  are  used,  the  vertical  angle  lines  are  run  at  intervals  of  4 
or  5  miles  in  one  direction,  while  roads  at  intervals  of  a  mile  are  run  in  the 
other  direction  with  aneroids,  checking  them  upon  the  crossings  of  the 
vertical  angle  lines.  Sketching  goes  on  coincidently  with  the  measurement 
of  heights. 


CHAPTER  V. 

SKETCHING. 

This,  being  by  far  the  most  important  part  of  the  work  of  map 
making,  should  be  done  by  the  most  competent  man  for  this  work  in  the 
party — as  a  rule,  by  its  chief  Besides  the  fact  that  he  is  presumably  the 
best  sketcher  in  the  party,  there  is  another  reason  for  requiring  that  he 
should  execute  the  sketching.  He  is  held  responsible  for  the  quality  of 
the  work,  not  only  of  the  sketching,  but  also  of  the  accuracy  and  the 
sufficiency  of  the  control.  In  the  sketching  of  the  map  he  has  the  best 
possible  opportunity  for  examining  into  the  condition  of  the  control  and  of 
remedying  any  weaknesses. 

Upon  the  completion  of  the  secondary  triangulation,  the  traverse 
work,  and  the  measurement  of  heights  within  an  area,  which  may  be  lai-ge 
or  small  according  to  convenience — but  preferably  should  comprise  a  qiiarter 
sheet — ^he  should  cause  all  this  control  to  be  assembled  upon  one  sheet. 
The  traverse  lines  with  all  points  located  from  them  should  be  adjusted  to 
the  secondary  locations,  and  all  measurements  of  height  should  be  plotted 
upon  this  skeleton,  thus  presenting  in  complete  form  all  the  control  within 
the  area.  With  this  sheet  upon  a  sketching  board  the  chief  of  party 
should  go  over  the  ground,  sketching  the  di'ainage,  culture,  and  forms  of 
relief.  The  latter  should  be  sketched  in  actual  continuous  contours,  direct 
from  the  country  as  copy,  so  that  upon  leaving  the  sketching  stations  the 
only  work  remaining  to  complete  the  map  will  be  inking  and  lettering.  In 
heavy  country,  however,  where  the  contours  follow  one  another  closely,  it 
may  often  be  sufficient  to  put  in  on  the  stations  only  a  part  of  the  contours — 
every  fifth  one,  for  instance — in  order  to  economize  time  in  the  field. 
Stations  for  sketching  may  be  selected  with  the  utmost  freedom.  An  exact 
106 


SKETCHING.  107 

location  is  unnecessaiy.  Any  point  on  or  off  the  road  wliicli  affords  an 
ontlook  will  serve.  As  a  rule,  frequent  stations  should  be  made,  and  one 
should  not  attempt  to  sketch  at  great  distance  unless  the  conditions  are 
favorable,  as  they  may  be  in  a  country  of  large,  bold  featui'es.  It  may  be 
necessary  to  travel  over  all  the  roads  which  haA^e  been  traversed  and  to 
climb  many  hills  in  order  to  sketch  the  entire  area  satisfactorily.  On  the 
other  hand,  in  a  different  region  the  entire  area  may  be  sketched  by  a 
limited  amount  of  travel  or  from  a  few  elevated  points.  In  a  low  country 
of  small  features  much  travel  will  be  required,  as  these  details  must  be 
sketched  from  near  points.  In  a  bold  country  of  high  relief,  which  may 
be  sketched  entirely  from  a  few  points,  care  must  be  exercised  in  the 
selection  of  sketching  stations.  From  a  great  altitude  the  lower  details 
will  be  dwarfed  and  will  measurably  disappear,  while  from  low  points  the 
relations,  forms,  and  masses  of  the  greater  elevations  cannot  be  properly 
seen.  In  such  a  country  stations  at  different  elevations  must  be  selected  in 
order  to  see  all  parts  of  the  country  to  the  best  advantage.  The  extreme 
summits  will  prove  of  little  service  as  sketching  stations. 

Sketching-  is  artistic  work.  The  power  of  seeing  topographic  forms 
in  their  proper  shapes  and  proportions  and  of  transferring  these  impressions 
to  paper  faithfully  is  of  all  acquirements  one  of  the  most  difficult  to  obtain. 
The  difficulty  is  increased  by  the  necessity  of  expressing  form  by  means  of 
continuous  contour  lines  at  fixed  intervals.  This  work  involves  a  knowl- 
edge of  the  elements  of  structural  geology  and  good  judgment  in  applying 
them. 

Every  map,  whatever  its  scale,  is  a  reduction  from  nature  and  conse- 
quently must  be  more  or  less  generalized.  It  is  therefore  impossible  that 
any  map  can  be  an  accui'ate,  faithful  picture  of  the  country  it  represents. 
The  smaller  the  scale  the  higher  must  be  the  degree  of  generalization, 
and  the  farther  must  the  map  necessarily  depart  from  the  original. 

Now,  it  is  in  this  matter  of  generalization  that  the  judgment  of  the 
topographer  is  most  severely  tested.  He  must  be  able  to  take  a  broad  as 
well  as  a  detailed  vdew  of  the  country;  he  must  understand  the  meaning 
of  its  broad  features,  and  then  must  be  able  to  interpret  details  in  the  light 
of  those  features.     Thus,  and  thus  only,  will  he  be  competent  to  ma!^-^  iust 


108  A  MANUAL  OP  TOPOGEAPHIC  METHODS. 

generalizations.  This  will  enable  him  to  decide  what  details  should  be 
omitted  and  what  ones  preserved,  and,  where  details  are  omitted,  what  to 
put  in  their  places  in  order  to  bring  out  the  dominant  features. 

It  is  not  possible  to  define  the  degree  of  detail  which  the  maps  should 
represent.  The  limit  commonly  given — that  is,  the  limit  imposed  by  the 
scale  of  the  map — is  not  always  the  best.  In  representing  country  which 
has  little  plan  or  system,  such  as  moraines  or  sand  dunes,  it  is  well  to  work 
in  as  much  detail  as  the  scale  will  bear.  But  where  the  country  shows  a 
system  in  its  sti-ucture  to  which  the  minor  detail  is  subordinate,  the  omission 
of  some  of  this  detail  may  give  greater  prominence  to  the  larger  features. 
The  amount  of  detail  thus  omitted  must  necessarily  be  left  to  the  judgment 
of  the  topographer,  but  no  more  should  be  omitted  than  is  necessary  to 
give  full  expression  to  the  general  features  of  the  country. 

ORIGIN  OF  TOPOGRAPHIC  FEATURES. 

As  an  aid  in  the  interpretation  of  tlie  various  topographic  forms  which 
present  themselves,  the  following  brief  discussion  is  appended. 

Topographic  features  originate  from  a  variety  of  causes  and  are  modi- 
fied by  many  agencies.  They  are  formed  by  uplift  from  beneath,  of  great 
or  small  extent.  They  are  formed  by  deposition  from  volcanoes,  glaciers, 
water,  and  the  atmosphere.  They  are  formed  or  modified  by  aqueous  and 
ice  erosion.     They  are  modified  by  gravity. 

These  are  the  principal  agencies  in  producing  topographic  forms  as  we 
see  them  to-day.  These  forms  are  only  in  rare  cases  the  work  of  a  single 
one  of  the  above  agencies ;  generally  two  or  more  have  taken  part  in  pro- 
ducing the  present  condition.  Of  all  these,  aqueous  agencies  are  by  far 
the  most  potent.  Their  work  is  seen  in  nearly  all  topographic  forms,  while 
in  those  of  great  age  their  action  has  been  so  extensive  as  to  mask  or  oblit- 
erate all  supei-ficial  traces  of  the  action  of  any  other  agency. 


The  internal  stresses  of  the  earth,  however  produced,  have  resulted 
in  raising  certain  portions  of  the  crust  and  depressing  others.  Commonly 
these  movements  have  been  slow  and  of  srreat  duration.     Some  of  them 


OEIGIN  OP  TOPOGEAPHIC  FORMS.  109 

are  of  continental  extent,  producing  plateaus,  while  others  have  been  very 
limited  in  extent,  throwing  up  narrow  ridges  or  blocks.  They  have 
uplifted  the  strata  at  various  angles,  so  high  in  some  cases  as  to  throw  them 
beyond  the  vertical,  infolding  the  strata  and  even  breaking  them  by  faults. 

Incidental  to  the  uplifts  are  flexures  and  faults.  The  flexures  may  be 
classed  as  anticlinal  folds,  where  they  are  bent  downward  on  either  side, 
and  monoclinal  flexures,  where  local  strata  first  bend  downward  and  then 
by  a  reverse  curve  resume  horizontality.  In  a  fault  the  rock  is  divided  by 
a  fracture  and  one  part  is  moved  up  past  the  other. 

It  is  through  uplift  that  continuous  mountain  ranges,  ridges,  and 
inclined  plateaus  have  originated — not,  howcA'er,  in  the  shapes  that  appear 
to-day,  for  most  of  them  during  and  since  their  rise  have  been  carved  by 
erosion  out  of  all  resemblance  to  the  forms  which  uplift  alone  would  have 
given  them. 

The  ridges  and  valleys  of  the  Appalachian  region  are  the  results  of 
uplifts,  with  numerous  sharp  folds  and  faults,  which  raised  at  various  angles 
an  alternation  of  hard  and  soft  beds,  from  which  erosion  has  since  carved 
the  existing  alternations  of  ridge  and  valley. 

Other  movements  of  uplift,  resulting  from  the  intrusion  among  the 
strata  of  great  lenses  of  volcanic  rock,  have  usually  resulted  in  the  forma- 
tion of  elliptic  mountains  or  groups  of  mountains.  As  these  movements 
have  occurred  at  different  periods  in  geologic  history,  some  have  been 
affected  more,  others  less,  by  erosion.  Certain  mountains  of  this  volcanic 
type  present  to-day  an  aspect  little  affected  by  erosion,  while  others  have 
been  greatly  modified  by  its  agency. 

Sierra  la  Sal,  in  eastern  Utah,  is  an  example  of  this  class.  Here  the 
stratified  beds  above  the  volcanic  rock  which  were  bent  upward  by  the 
uplift  were  probably  broken  over  the  top,  and  have  been  removed  by 
erosion  until  now  they  only  sm-round  the  base  of  the  group,  dipping  away 
from  it  steeply,  forming  hogbacks. 

In  New  Mexico  there  are  seen  numerous  volcanic  "necks"  rising 
abruptly  from  the  plateau.  These  necks  are  intrusions  of  volcanic  rocks, 
which  were  forced  up  while  molten  into  the  stratified  rocks.  The  latter 
have  since  been  eroded  away,  leaving  the  harder  necks  as  isolated,  prepip- 
itous  mountains. 


110  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

DEPOSITION   FROM   A'OLCANIC   ACTION. 

Deposits  from  volcanic  action  may  be  grouped  as  follows:  (1)  of  liqviid 
lava,  in  tlie  forms  a,  of  streams  and  lakes,  resulting  in  plains,  tables,  and 
mesas,  and  h,  of  cones  with  craters,  with  gentle  slopes,  (2)  of  scoriae  and 
cinders,  of  which  have  been  built  cones  with  steep  slopes,  either  with  round 
tops  or  with  craters. 

Deposits  of  the  first  group  consist  largely  of  fields"  of  Ijasalt  which  have 
been  poured  out  from  low  vents  or  craters  and  spread  in  horizontal  sheets, 
in  many  cases  covering  great  extents  of  territory.  The  Snake  river  plains 
of  Idaho  furnish  an  example.  As  most  of  these  eruptions  are  of  recent  date, 
these  sheets  of  basalt  have  suffered  little  from  erosion,  then-  form  remaining 
much  the  same  as  when  they  were  pom-ed  out  and  spread  over  the  land. 
The  surface  is  undulating,  broken  here  and  there  by  low  cliffs  marking  the 
edges  of  the  flow,  and  by  cracks  and  fissm-es  here  and  there,  especially  near 
the  borders  of  the  field.  Owing  to  the  frequency  of  the  fissures,  flowing 
water  is  scarce  upon  these  basalt  fields,  for  the  streams,  sinking  in  the  fissures, 
find  undergi'ouud  channels,  to  reappear  at  the  borders  of  the  fields  in  springs. 

AQUEOUS   AGENCIES. 


The  principal  agency  in  shaping  topographic  forms  is  aqueous  erosion. 
In  nine-tenths  of  the  area  of  the  United  States  the  work  of  this  agency  is 
prominent,  while  over  miich  the  larger  part  of  the  country  the  forms  are 
apparently  due  entirely  to  this  action.  It  is  so  commonly  seen,  that  the 
topographer  finds  himself  unconsciously  reasoning  in  accordance  with  its 
laws  and  attempting  to  apply  them  to  forms  produced  by  other  agencies. 
A  country  shaped  by  aqueous  erosion  is  to  him  a  "  regular"  country,  while 
one  shaped  by  other  agencies,  less  known,  is  iiTegular.  The  foi-mer  can,  to 
some  extent,  be  foreseen.  In  such  a  region,  one  reasons  from  the  seen  to 
the  unseen,  while  the  vagaries  of  the  latter  can  seldom  be  predicted.  By 
its  agency  the  Appalachian  mountains  have  been  reduced  from  a  compli- 
cated system  of  mountain  folds  to  the  present  comparatively  low  and  simple 
system  of  sandstone  ridges  and  limestone  valleys.     In  the  Cumberland 


OEIGIN  OF  TOPOGEAPHIC  FOEMS.  1 1 1 

plateau  has  been  produced  its  remarkably  complex  drainage  system.  From 
enormous  plateaus  have  been  carved  the  great  ranges  of  Colorado,  with 
their  peaks,  canyons,  and  clififs.  From  the  plateaus  of  the  Colorado  drain- 
age system  thousands  of  feet  of  rock  have  been  worn  away,  leaving  here 
and  there  great  cliffs  and  high  plateaus  to  show  the  magnitude  of  its  work, 
while  the  great  canyons  dividing  the  lower  plateaus,  some  of  them  a  mile  in 
depth,  though  the  least  among  its  works,  are  the  topographic  wonders  of 
the  world.  From  the  moment  the  land  rose  above  the  sea,  this  agency  of 
destruction  has  been  at  work,  and  its  labors  will  not  cease  until  the  land 
again  sinks  beneath  the  waves. 

The  action  of  water  on  rocks  may  be  divided  into  three  parts — weather- 
ing, transportation,  and  corrasion.  The  rocks  of  the  general  surface  of  the 
land,  or  the  terrain,  are  disintegrated  and  converted  into  soil  by  weathering. 
The  material  thus  loosened  is  transported  by  streams,  and  while  thus  being 
transported  it  helps  to  corrade  other  material  from  the  channels  of  the 
streams.  In  weathering,  the  chief  agents  are  solution  by  water,  frost,  the 
mechanical  beating  of  rain,  gravity,  and  vegetation.  Some  rocks,  particu- 
larly limestones,  are  entirely  dissolved  by  water,  especially  when  it  is  charged 
with  carbonic  acid ;  others  are  dissolved  only  in  part  and  the  remaining  part 
is  thus  disintegrated.  Rocks  are  cracked  and  broken  by  the  freezing  of 
water  in  their  interstices.  When  the  foot  of  a  cliff  is  undermined  by  erosion, 
the  upper  portion,  failing  of  support,  breaks  off  in  fragments  by  its  own 
weight.  The  roots  of  plants  pushing  their  way  into  the  interstices  of  rocks 
pry  them  apart  and  thus  aid  in  disintegration.  In  general,  soft  rocks  disin- 
tegrate more  rapidly  than  hard  rocks  and  soluble  rocks  more  rapidly  than 
insoluble  rocks.     Disintegration  is  more  rapid  in  a  moist  than  in  a  diy  climate. 

The  product  of  disintegration  is  soil,  and  this  may  be  regarded  in  future 
discussion  as  a  soft  bed  subject  to  the  same  laws  of  corrasion  and  transpca-- 
tation  as  oth,er  beds,  with  only  such  modifications  as  its  want  of  cohesion 
requires. 

TRANSPORTATION  AND  CORRASION. 

Rain  falls  upon  the  surface,  a  portion  of  it  sinks  and  reappears  in  springs, 
while  another  portion  flows  down  the  surface  and  collects  in  water  courses, 
which,  joining  one  another,  produce,  finally,  large  streams.     During  a  rain 


112  A  MxiNUAL  OF  TOPOGEAPHIC  METHODS. 

storm  the  entire  surface  is  a  network  of  water  courses,  from  the  most  minute 
rills  to  the  main  streams,  and  in  studying  transportation  and  corrasion  the 
action  of  these  minute  rills,  which  cover  the  entire  terrain,  must  be  considered 
as  fully  as  that  of  the  main  stream  and  its  primary  branches. 

Con-asion  is  effected  by  the  detritus  which  running  water  holds  in 
suspension.  Soft  rocks  are  corraded  rapidly,  hard  rocks  slowly.  The  rate 
of  corrasion  is  increased  by  an  increase  in  the  volume  of  the  stream,  an 
increase  in  its  velocity,  an  increase  in  the  amount  of  detritus  borne  by  it, 
and  by  increased  coarseness  of  that  detritus.  Hence  it  is  that  the  tiny  rain- 
water rivulets  have  very  feeble  corrasive  powers;  but  as  they  combine  into 
larger  and  larger  streams,  and  as  they  wash  into  their  channels  a  larger  and 
larger  amount  of  detritus,  and  as  the  slope  of  their  beds  becomes  greater, 
their  power  for  corrading  their  beds  increases,  and  hence  it  is  that  the  cor- 
rading  power  of  the  main  stream  is  greater  than  that  of  any  of  its  branches, 
and  in  the  main  stream,  if  the  slope  were  uniform,  the  corrasive  power 
would  be  greatest  near  its  mouth. 

Suppose  a  stream  to  have  initially  a  uniform  slope  from  its  source  to 
its  mouth — then  its  volume,  its  velocity,  and  the  amount  of  detritus  borne 
by  it  will  be  greatest  near  its  mouth;  and  corrasion,  although  going  on  all 
along  its  course,  will  be  most  rapid  there.  The  slope  of  the  stream  will 
therefore  be  reduced  most  rapidly  in  the  lower  part  of  its  com-se,  and  thence 
progressively  up  stream.  It  results  from  this  that  the  normal  profile  of  a 
stream  bed  is  a  cm-ve,  concave  upward. 

While  the  slope  of  the  stream  bed  remains  considerable  and  the  velocity 
consequently  great,  the  stream  flows  in  a  comparatively  straight  channel, 
and  devotes  its  energies  to  deepening  its  bed,  and  thus  reducing  its  slope. 
As  the  slope  becomes  thus  reduced  the  course  of  the  stream  changes  to  a 
crooked,  winding  one,  and  its  corrasive  energies  are  diverted  from  its  bottom 
to  the  sides  of  its  bed.     It  is  then  said  to  approach  "baseleveL" 

Swift  streams  commonly  flow  in  straight-  channels;  sluggish  streams, 
in  crooked  channels. 

While  lowering  its  bed  by  corrasion  the  main  stream  lowers,  necessarily, 
the  mouths  of  its  immediate  affluents,  and  these  affluents  are,  therefore,  in 
addition  to  their  own  proper  work,  obliged  to  cut  their  lower  courses  down 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII.  PL.  Vlll. 


A  BIT  OF  THE  GREAT  PLAINS,  COLO.,  AND  KAN  ,  NEAR  BASE  LEVEL. 


Scale    125,000 
ContoTxr  Irrteirv-al  2 5  feet 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII.     PL.  IX. 


A  BIT  OF  THE  ATLANTIC  PLAIN,  VA.    NEAR  BASE  LEVEL. 

Scale    125,000 
Contour  later-T-al  50    feet 


ORIGm  OF  TOPOGRAPHIC  FORMS.  Ii3 

to  a  level  with  the  main  stream.  The  same  operation  which  is  going  on  in 
the  main  stream  is  going  on  in  these  affluents,  but  with  different  intensity, 
owino-  to  their  smaller  volume  of  water  and  perhaps  smaller  amount  of  sedi- 
ment, and  to  the  fact  noted  below,  that  their  mouths  are  constantly  being- 
lowered.  Now,  following  up  these  branches  as  they  subdivide  into  smaller 
and  smaller  streams,  a  point  is  finally  reached  where  the  little  rivulets,  with 
their  feeble  cutting  power,  are  only  able  to  keep  their  lower  courses  cut 
down  to  the  level  of  the  stream  to  which  they  are  tributary.  They  have 
no  energy  to  spare  in  working  back  up  their  own  courses.  At  this  point  the 
curve  changes  from  one  concave  upward  to  one  convex  upward.  This  con- 
vex curve  is  the  curve  of  all  the  minor  rain-water  rivulets — in  short,  it  is  the 
curve  of  the  terrain — while  the  concave  curve  is  the  curve  of  the  water 
courses.  The  former  is  the  curve  of  the  upper  relief  of  the  country,  the 
latter  is  the  curve  of  the  valleys. 

The  relative  extent  of  these  two  curves  depends  mainly  upon  the 
climate  and  upon  the  range  of  elevation  of  the  country,  or,  in  other  words, 
upon  the  relative  rapidity  of  corrasion  of  their  beds  by  the  perennial  streams, 
and  the  erosion  of  the  teiTain  by  the  rain-water  rivulets.  In  a  well- watered 
reo-ion,  where  the  range  of  elevation  is  small,  and  where  the  larger  streams 
are  near  base  level,  the  hill  forms  are  broad,  rounded,  and  convex,  and  the 
valleys  are  equally  rounded,  with  concave  forms.  Of  this  type  is  the  undu- 
lating billowy  surface  of  the  Grreat  Plains  and  the  Atlantic  and  Gulf  plains 
of  the  Southern  states. 

Where  the  range  of  elevation  is  great,  the  curves  both  of  valley  and 
ridge  become  sharper  and  more  angular.  The  streams  have  a  greater  fall 
and  proportionally  increased  power,  and  therefore  cut  more  rapidly;  but, 
on  the  other  hand,  they  have  more  work  to  perform.  The  Cumberland 
plateau,  with  its  intricate  network  of  streams,  consists  of  a  close  alternation 
of  ridges  and  valleys,  with  straight  slopes  at  very  steep  angles,  the  bottoms 
of  the  gorges  and  the  summits  of  the  ridges  being  but  slightly  rounded. 
Few  of  the  streams  have  reached  base  level,  except  in  some  cases  near  their 
mouths,  and  corrasion  of  their  beds  is  still  active.  In  a  high  mountain  range 
all  these  features  become  still  more  accented.  The  main  streams  have  a 
steep  descent  and  corrade  their  beds  rapidly.  Their  valleys  are  narrow, 
MON  xxii 8 


114  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

with  steep  slopes  on  both  sides.  The  mouths  of  the  secondary  streams  are 
rapidly  lowered,  and  thereby  their  work  is  greatly  increased. 

There  is  therefore  a  distinction  to  be  observed  between  superficial 
erosion  or  erosion  by  the  petty  rain-water  streams  on  the  one  hand  and 
that  by  the  larger  streams  on  the  other.  The  first  forms,  as  a  rule,  convex 
slopes;  the  last,  concave  slopes.  Between  them,  however,  no  sharp  line  can 
be  drawn.  In  general,  the  former  erodes  soil  only,  the  soft  superficial  bed, 
while  the  latter,  if  swift,  is  at  work  chiefly  on  rock.  The  energy  of  the 
former  is  widely  dispersed,  that  of  the  latter  is  concentrated.  The  general 
reduction  of  the  surface  is  done  by  the  former,  while  the  latter  is  confined 
to  deepening  narrow  stream  beds.  Where  the  main  streams  are  near  base 
level,  superficial  erosion  goes  on  more  rapidly  than  stream  corrasiou,  since 
the  slope  and  velocity  of  the  streams  are  near  a  minimum.  Where  the 
streams  are  still  corrading  rapidly,  their  beds  are  usually  lowered  faster 
than  the  terrain,  and  the  balance  is  more  and  more  on  the  side  of  the 
streams,  the  greater  the  range  of  elevation.  In  a  mountain  region,  as  has 
just  been  stated,  the  gorges  are  cut  far  below  the  spurs  and  summits. 
Hence,  where  stream  corrasion  predominates  over  surface  erosion,  the  con- 
cave curve  predominates,  and  where  surface  erosion  is  more  rapid  than  cor- 
rasion by  the  streams,  the  convex  curve  is  the  ruling  one. 

In  an  arid  regioia,  where  the  rain-fall  is  not  only  scanty,  but  spasmodic 
in  character,  coming  mainly  in  sudden  showers  of  great  volume,  but  short 
duration,  the  stream  beds  are  few  in  number.  The  drainage  system  is 
scanty  and  imperfectly  developed.  The  weathering  of  rocks  goes  on  slowly, 
and  consequently  the  soil  bed  is  thin.  The  soft  material  which  the 
streamlets  can  erode  is  not  abundant.  Consequently  the  scanty  rains  do 
little  surface'  erosion,  but  as  they  collect  in  large  volume  in  the  few  water 
courses,  they  deepen  them  at  a  rapid  rate.  Erosion  of  the  terrain  in  an  arid 
region  is  therefore  slow,  while  stream  corrasion  is  proportionally  rapid. 

It  is  frequently  the  case  that  streams  collect  their  waters  from  high 
mountains,  and  on  their  way  to  the  sea  pass  down  through  arid  regions. 
The  action  of  such  streams  upon  the  arid  region  is  the  same  as  above 
described  from  streams  originating  within  this  region,  except  that  it  is  more 
intense.     Little  or  none  of  the  waters  of  such  a  stream  flows  over  the  ter- 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII.     PL.  X. 


A  PORTION  OF  THE  CUMBERLAND  PLATEAU,    IN   W.  VA. 


Scale    125X100 
CoiLtoiar  Interval  100  feet 


U.  S.  GEOLOGICAL  SURVEY, 


MONOGRAPH  XXII.  PL.  XI. 


CANYONS  IN  HOMOGENEOUS  ROCKS. 


Scale    125,000 
CoiLto-ur  IiiteTrv-al25  feet 


ORIGIN  OF  TOEOGRAPHIC  FORMS.  1 15 

rain  of  the  arid  area,  to  contribute  to  the  planing  down  of  its  surface ;  but, 
on  the  other  hand,  the  vokime  and  consequently  the  energies  of  the  stream 
for  corrasion  are  greatly  increased  by  the  copious  contributions  from  the 
mountain  region.  Therefore,  in  such  cases  corrasion  by  the  streams  reaches 
a  maximum,  relative  to  erosion  of  the  terrain. 

It  is  tluTS  that  canyons  in  the  arid  region  are  formed.  They  are  found 
wherever,  from  any  cause,  stream  corrasion  is  decidedly  more  rapid  than 
surface  erosion. 

Such  canyons,  when  in  homogeneous  rocks,  rarely  contain  vertical 
cliffs.  These  are  commonly  formed  in  strata  of  differing  hardness  by  sap- 
ping and  undermining,  which  will  be  explained  later. 

In  certain  parts  of  the  arid  region,  notably  in  the  Great  basin,  the  rain- 
fall is  so  scanty  that  the  drainage  systems  are  very  feeble.  The  little  rain 
that  falls  in  the  valleys  is  at  once  absorbed  by  the  thu-sty  soil  or  the  atmos- 
phere, while  the  streams  which  flow  down  from  the  mountains,  cutting,  it 
may  be,  deep  canyons  in  their  sides,  dwindle  away  on  reaching  the  valley, 
depositing,  as  they  sink,  their  loads  of  detritus.  With  this  detritus  have 
been  floored  to  a  vast  depth  most  of  the  valleys  of  the  Great  basin.  It  has 
been  deposited  there,  instead  of  being  carried  off  to  the  sea.  The  Great 
basin,  which  is  in  reality  a  large  number  of  basins  more  or  less  independent 
of  one  another,  is  without  outlet  simply  because  of  its  small  rainfall.  Were 
the  rainfall  to  increase,  it  would  soon  contain  many  lakes,  and  as  the  water 
rose  these  would  overfow,  the  higher  flowing  into  the  lower  and  the  lower 
flowing  into  the  sea.  The  streams  connecting  them  and  the  sea,  would  soon 
corrade  channels,  cutting  them  down  to  lower  and  still  lower  levels,  and 
progressively  draining  these  lakes,  and  thus  a  di'ainage  system  would  be 
established. 

^nks  exist  in  other  parts  of  the  country,  but  are  there  due  to  different 
causes.  They  are  common  in  the  Appalachian  region.  In  these  sinks  the 
water  has  an  undergi'ound  outlet  through  passages  in  the  soluble  limestone 
with  which  the  valleys  are  floored.  They  are  common  among  the  terminal 
moraines  of  the  continental  glacier,  in  Minnesota,  Wisconsin,  Michigan,  and 
New  England,  where  they  are  called  kettles.  Here  glacial  material  has  been 
deposited  so  recently  that  time  has  not  yet  been  afforded  for  the  establish- 
ment of  drainage  systems. 


116  A  MANUAL  OF  TOrOG-KAPHIC  METHODS. 

Every  stream  tends  to  extend  its  drainage  area  Ly  erosion  at  its 
sources  on  all  sides,  necessarily  at  the  expense  of  its  neighbors.  The  stream 
having  the  most  rapid  fall  erodes  the  margin  of  its  basin  most  rapidly. 
Hence  in  their  struggle  for  existence  the  stream  having'  the  most  rapid  descent 
succeeds  in  drawing  area  from  others.  But  in  so  doing  it  diminishes  its  own 
rate  of  fall,  so  that  eventually  a  state  of  equilibrium  among  streams  may  be 
reached.  This  extension  of  basins  is  called  piracy.  It  is  going  on  actively 
in  the  Appalachian  valley,  Avhere  numerous  examples  may  be  found. 

AVhile  under  certain  circumstances  the  courses  of  streams  are  mutable, 
under  other  conditions  streams  maintain  their  courses  with  gi-eat  pertinacity. 
Of  this,  water  gaps  and  canyons  across  mountain  ranges  are  striking  results. 
Where  such  a  canyon  is  found,  the  river  flowed  before  the  range  or  ridge 
existed.  The  range  may  have  risen  across  its  course,  in  which  case  the 
river,  like  a  circular  saw,  maintained  its  course  by  corrasion,  cutting  the  can- 
yon as  the  mountain  rose.  Of  this  action  the  canyon  of  Green  river  through 
the  Uinta  range  is  an  example. 

Or,  the  river,  draining  a  surface  of  soft  or  soluble  rocks,  and  eroding 
this  surface  down,  may  have  uncovered  a  ridge  of  hard  rock  lying-  across 
its  course.  In  this  case,  like  the  other,  the  river  maintains  its  course  by 
cutting  a  canyon  through  the  ridge.  The  Appalachian  valley  presents  num- 
berless examples  of  water  gaps  formed  as  above  described.  Among  them 
maybe  mentioned  Delaware  Water  gap,  through  which  Delaware  river  passes 
Kittatinn}^  mountain,  gaps  of  tiie  Susquehanna  and  the  Juniata,  that  of  the 
Potomac  at  Harpers  Ferry,  and  Big  Moccasin  gap,  while  Little  Moccasin 
gap  is  in  process  of  completion.  While  these  are  prominent  and  well  known 
cases,  in  certain  localities,  every  little  ridge  is  cut  into  a  line  of  knobs  by 
them,  so  that,  next  to  the  parallelism  of  its  ridges  and  valleys,  the  water  gaps 
of  the  Appalachian  valley  constitute  its  most  prominent  feature.  S%ich  of 
these  gaps  as  can  be  shown  should  appear  on  the  map,  and  when  owing  to 
the  minuteness  of  these  features  it  becomes  necessary  to  omit  them,  one 
should  recognize  the  fact  that  the  formation  in  this  region  is  that  of  parallel 
ridges  and  so  represent  the  structure. 

Wind  gaps  are  abandoned  water  gaps,  from  which  the  stream  has 
been  drawn  away  by  a  more  powerful  neighbor.     These  should  not  be 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII.     PL.  XII. 


CANYONS  AND  CLIFFS  IN   ROCKS  NOT  HOMOGENOUS,   N.  M. 

Scale     125.000 
ContoiiT-  liXtei-val  50  feet 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII-  PL.  XIII- 


y'<. 


A  PORTION  OF  THE  GRAND  CANYON  OF  COLORADO  RIVER,  ARIZ- 

Scale      20O.000 
ConLour    Interval     250     feet 


ORIGIIf  OF  TOPOGRAPHIC  FORMS.  117 

confounded  with  passes,  or  low  points  in  mountain  rang-es,  caused  bj  the 
eating  away  of  divides  at  the  heads  of  streams. 

The  valley  of  every  stream  above  base  level  slopes  not  only  toward 
the  stream,  but  with  it — i.  e.,  toward  its  mouth.  Every  branch  on  entering 
the  valley  feels  the  influence  of  this  slope  and  turns  its  course  in  greater 
or  less  deg'ree  down  tli^  valley,,  entering  the  main  stream  at  an  acute  angle. 
Similarly  the  main  stream  feels  the  influence  of  the  tributary  and  turns  toward 
it;  hence  the  tributary  commonly  joins  the  main  stream  at  the  head  of  a  bend 
in  the  latter. 

When,  however,  a  stream  has  recently,  by  the  extension  of  its  drain- 
age basin,  tapped  an  adjacent  stream,  the  stream  so  tapped  may  not  yet 
have  accommodated  its  course  to  that  of  the  principal  stream,  so  that  it  still 
enters  it  at  an  obtuse  angle. 

Again,  when  the  stream  is  near  base  level  a  different  condition  is  pre- 
sented. The  main  stream  is  on  a  ridge  of  its  own  construction,  and  the 
tributary  often  comes  into  the  valley  at  a  lower  level  than  the  ridge  and 
flows  parallel  with  it  for  a  distance  before  breaking  through  and  joining  its 
waters.  Loup  fork  of  the  Platte  river,  Nebraska,  is  an  example  of  this. 
The  Platte  flows  there  upon  a  ridge  of  its  own  creation.  The  Loup  comes 
down  into  its  valley  and  flows  parallel  to  it  for  many  railes. 

As  was  stated  before,  a  stream  near  base  level  becomes  crooked  and 
winding.  It  has  ceased  to  corrade  its  bottom,  but  coiTades  the  sides  of  its 
bed,  especially  at  the  heads  of  its  bends,  and  deposits  its  load  on  the  inside 
of  its  bends.  Its  course  changes  frequently,  now  extending  its  bends 
farther  into  the  bank  and  now  cutting  them  off.  In  this  way  it  eventuallv 
excavates  a  bi'oad  alluvial  bottom,  which  may  be  subject  to  overflow  when 
the  stream  is  in  flood  and  through  which  the  stream  Avinds  in  long  curves, 
of  size  roughly  proportional  to  the  magnitude  of  the  stream. 

In  the  preceding  pages  no  reference  has  been  made  to  the  influence  of 
structure  upon  topographic  forms.  The  alternation  of  hard  and  soft  beds 
of  rock  and  the  dip  of  these  beds  have  decided  influence  upon  topographic 
forms,  which  are  now  to  be  considered.  The  influence  of  these  factors  upon 
topography  is,  it  must  be  premised,  greater  in  the  arid  regions  of  the  West 
than  in  the  moister  East.     The  reason  of  this  is  that  disintegration  is  much 


118  A  majSiual  of  topographic  methods. 

more  rapid  in  the  moister  climate,  and  consequently  that,  finding  an 
abundance  of  material  in  the  bed  of  soil,  a  larger  proportion  of  the  ener- 
gies of  corrasion  are  devoted  to  removing  it,  while  proportionately  less  is 
deA^oted  to  rock  work.  Still  the  effect  of  structure  is  by  no  means,  absent 
in  the  East. 

Since  disintegration  and  corrasion  of  hard  or  'insoluble  rocks  go  on 
slowly,  and  of  soft  or  soluble  rocks  rapidly,  the  elevated  areas  are  conse- 
quently, as  a  rule,  composed  of  the  former,  while  the  depressed  areas 'are 
commonly  of  the  latter  class  of  rocks.     It  is  the  survival  of  the  hardest. 

When  erosion  has  left  a  peak,  a  projection,  spur  or  boss,  a  butte  or 
mesa,  a  neck  or  dike,  it  is  commonly  because  the  material  is  harder  than 
that  adjoining.  The  valleys  of  the  Appalachian  region  are  almost  without 
exception  cut  in  soluble  limestone,  while  the  ridges  are  mainly,  and  the 
higher  ones  entirely,  of  sandstone. 

Streams  usually  make  their  channels  along  lines  of  least  resistance. 
They  accommodate  themselves  to  the  softness  of  the  rocks  and  avoid 
obstacles.  The  more  rapid  the  stream,  however,  the  less  does  it  care  for 
obstacles,  while  gentle  streams  are  most  easily  diverted. 

The  level  surface  of  a  plateau  is  generally  the  summit  of  a  hard  bed, 
from  which,  it  may  be,  softer  beds  have  been  washed  away  and  on  which 
erosion  has  comparatively  come  to  a  standstill. 

Where  rocks  of  different  hardness  are  subjected  for  the  same  time  to 
an  equal  intensity  of  corrasion,  since  the  effect  upon  the  softer  rock  is 
greater  than  that  upon  the  harder,  it  will  be  brought  down  to  gentler 
slopes;  in  other  words,  other  things  being  equal,  the  harder  the  rock  the 
steeper  the  slope,  the  softer  the  rock  the  more  gentle  the  slope.  Now,  let 
this  proposition  be  applied  to  the  cross  sections  of  stream  beds.  Suppose 
two  stream  beds,  one  in  soft  rock,  another  in  hard  rock,  both  of  them  sab- 
iected  to  the  same  climatic  agencies  and  the  same  corrasive  action  for  the 
same  time.  In  these  two  rocks  the  stream  beds  will  be  carved  somewhat  as 
shown  in  Nos.  1  and  2,  in  Figure  13,  indicating  progressive  stages  of  opera- 
tion. 

The  simplest  case  for  consideration  and  a  very  common  one  is  that  of 
horizontal  beds,  alternately  hard  and  soft,  such  as  are  represented  in  Fig- 


U.  8.  GEOLOGICAL  SURVEY, 


MONOGRAPH  XXII.     PL.  XIV. 


WATERGAPS,  PA. 

Scale    es.ioo 
Contour  Interval  20  feet 


ORIGIN  OF  TOPOGEAPHIC  FORMS. 


119 


ure  13,  Nos.  3  and  4  Suppose  No.  3  to  represent  a  cross  section  of  a  canyon, 
the  upper  bed  of  tlie  plateau  being  hard,  succeeded  by  soft  and  hard  beds 
in  alternation,  as  is  seen  in  the  Grand 
canyon  of  the  Colorado,  PL  xiii.  The 
course  of  the  stream  in  forming  this 
canyon  is  shown  by  the  light  lines  in 
the  figure.  It  cuts  first  a  canyon  with 
steep  sides  in  the  upper  hard  bed, 
an  operation  which  perhaps  consumes 
much  time.  Then  reaching  the  softer 
bed  below,  it  bu.rrows  rapidly  into  it, 
at  the  same  time  undermining  the  bed 
above,  which  from  its  weight  breaks 
away,  leaving  cliffs.  A  similar  opera- 
tion carries  it  through  the  next  hard 
and  soft  beds.  Thus  a  succession  of 
cliffs  and  terraces  is  formed.  The 
presence  of  cliffs  in  a  canyon  wall 
generally  indicates  that  the  bed  be- 
neath the  cliff  is  more  easily  eroded  fig.  i3— .cross  sections  of  canyons. 
than  that  above  it.  The  fragments  of  the  cliff  falling  upon  the  slope  of  the 
soft  bed  below  form  what  is  known  as  a  talus. 

The  above  is  a  common  case  in  a  plateau  region,  since  the  surface  bed 
is  usually  hard.  Where  the  surface  consists  of  a  soft  bed.  No.  4,  Fig.  13, 
represents  the  condition  of  the  canyon  walls.  The  undulating  surface  of 
the  soft  bed  slopes  down  to  the  cliff  produced  by  undermining  the  hard  bed 
beneath.     Otherwise  the  case  is  similar  to  that  described  above. 

■  A  third  case  is  afforded  by  the  Black  canyon  of  the  Gunnison  in  Col- 
orado, where  a  hard  sandstone  forms  the  surface  of  the  plateau,  underlain 
by  granite.  A  section  is  represented  by  No.  6  in  Fig.  13.  The  sandstone 
stands  at  an  angle  of  about  30°,  beneath  which  are  the  walls  of  the  granite 
canyon,  which  are  somewhat  steeper,  the  angle  of  slope  being  perhaps  40° 
to  45°.  There  is  no  undermining  and  consequently  4here  are  no  vertical 
cliffs. 


120 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


No.  2. 
Fig.  14. — Cross  sections  i 


I  inclinerl  "beds. 


Consider  next  the  case  of  a  stream  flowing  parallel  to  the  strike  of 
inclined  beds,  where  they  are  alternately  hard  and  soft.  When  the  incli- 
nation of  the  beds  is  not  great,  the  stream,  having  cut  down  to  the  surface 

I A  of  the  hard'  bed,  as  represented  in 
No;  1,  Fig.  14,  tends  to  travel  later- 
ally down  the  dip  of  the  bed,  under- 
mining both  soft  and  hard  beds  on 
the  lower  side  and  extending  the  slope 
on  the  upper  side.  When  the  dip  is 
considerable,  it  may  carry  away  all 
the  material  on  the  upper  side,  as 
in  No.  2,  Fig.  14 

In  this  way  streams  may  cut  broad 
swaths  across  the  terrain  and  remove 
both  hard  and  soft  beds  from  great 
areas  of  inclined  plateaus. 
Fine  examples  of  streams  flowing  on  the  strike  of  hard  inclined  strata 
are  seen  in  the  hogbacks  of  Colorado. 

Next,  consider  the  longitudinal  profile  of  a  stream  which  is  cutting  its 
bed,  when  flowing-  over  a  succession  of- beds  alternately  hard  and  soft. 
Since  it  cuts  soft  rocks  more  rapidly  than  hard  ones,  its  profile  will  show 
irregularities.  Wliere  flowing  over  soft  beds,  its  current  is  less  rapid  than 
over  hard  beds  of  rock.  The  stream  adjusts  its  proflle  to  the  work  to  be 
performed. 

The  ultimate  result  of  aqueous  erosion  upon  a  surface  is  to  reduce  it 
to  a  plain  of  slight  elevation,  of  gentle,  easy  slopes.  It  then  approaches 
base  level,  a  condition  where  the  entire  surface  resembles  the  condition  of 
a  base-level  stream,  where  vertical  coiTasion  is  practically  at  an  end.  Abso- 
lute base  level  is  a  conception  merely,  which  many  regions  approach,  but, 
owing  to  the  fact  that  as  the  slopes  become  gentler,  erosion  becomes  feebler, 
they  cannot  reach. 

The  stage  of  progress  of  an  area  toward  base  level  is  said  to  indicate 
its  age.  In  youth  it  may  present  a  great  elevation  and  high  relief.  Its 
streams  may  have  rapid  courses  with  irregular  profiles,  broken  by  lakes. 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXU.     PL.  XV, 


THE  RIDGE  OF   MISSISSIPPI   RIVER,  LA. 


Scale  62,5oo 
Contoirr  IiLt-erval  5  feet 


ORIGIN  OF  TOPOGEAPHIC  FORMS.  121 

rapids,  and  falls.  As  the  age  of  the  region  increases  these  inequalities  are 
cut  away.  The  lakes  are  drained,  the  falls  and  rapids  disajDpear.  The 
mountains  and  hills  are  worn  down,  and  finally  the  entire  surface  is  reduced 
to  a  low  rolling  expanse.  The  region  approaches  base  level.  It  is  in  its 
old  age.     Plains  represent  old  age  among  topographic  featm-es. 

The  life  of  a  topographic  area  is  not  to  be  measured  in  years,  but  in 
its  cycle  of  changes,  which  have  little  reference  to  time.  The  time  required 
to  run  through  its  life  differs  with  the  conditions  under  which  and  the  ma- 
terials upon  which  erosion  acts.  It  varies  with  the  intensity  of  erosive 
action  and  with  the  amount  of  work  to  be  done. 

Sometimes  a  region  after  being  reduced  nearly  to  base  level  has  been 
again  elevated.  Such  elevation  brings  again  into  action  the  erosive  agen- 
cies to  carve  and  plane  the  terrain  a  second  time.  A  region  thus  restored 
to  .youth  by  elevation  is  the  mountain  region  of  North  Carolina.  The 
bench  level  of  the  country  is  an  old  base  level,  which  has  been  raised.  In 
this  the  streams  are  now  cutting  and  regulating  their  courses,  while  the 
bench  level,  in  its  gentle  undulations,  shows  the  old  base-level  sm-face, 
little  affected  as  yet  by  recent  erosion. 

DEPOSITION  FROM  WATER. 

When  the  swift  current  of  a  stream  is  checked,  as  by  a  reduction  of 
slope  or  by  a  widening  of  its  bed,  it  deposits  a  part  of  its  load.  It  is  thus 
that  river  banks,  river  and  lake  terraces,  and  bars  at  the  mouth  of  streams . 
are  made.  Of  the  building  of  river  banks,  fine  examples  are  seen  in  south- 
ern Louisiana.  Before  the  stream  was  lined  with  levees  the  Mississippi 
river  overflowed  its  banks  at  every  considerable  rise.  Loaded  with  detritus, 
it  suddenly  spread  over  its  banks  to  the  dimensions  of  an  inland  sea;  its 
velocity  was  thereby  checked  and  much  of  its  load  was  quickly  deposited, 
the  greater  part,  including  the  coarsest  material,  falling  on  its  immediate 
banks,  which  were  thereby  built  up  higher  than  the  adjoining  country.  The 
river  and  bayous  of  this  region  flow  on  the  tops  of  ridges  of  their  own  con- 
struction, the  only  land  above  the  swamps.  The  highest  ground  every- 
where is  that  on  the  immediate  river  bank,  whence  the  slope  is  away  from 
the  stream  on  either  hand  to  the  swamp,  as  shown  in  PL  xv. 


\ 


122  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

Now,  let  this  operation  be  extended  farther.  As  a  stream  builds  its 
ridge  higher  it  soon  reaches  a  condition  of  instability  and  it  then  forsakes  its 
bed  for  an  adjoining  lower  course.  It  builds  this  up  and  in  turn  abandons 
it.  So  in  time  it  builds  up  a  di-y  delta,  or,  as  it  is  called,  a  fan,  made  up  of 
a  radiating  group  of  abandoned  ridges  marking  its  former  courses. 

Lake  terraces  are  formed  by  the  collection  of  material  at  the  water's 
edge.  Whether  brought  down  by  gravity  alone  or  transported  by  water, 
its  descent  is  checked  on  reaching  the  water  and  it  accumulates  at  the 
water's  edge. 

GLACIAL  DEPOSITION, 

The  northern  part  of  the  United  States  was,  in  recent  geologic  times, 
covered  by  a  sheet  of  ice,  a  glacier  of  continental  dirnensions.  Its  bound- 
aries, within  the  United  States,  included  New  England,  New  York,  north- 
ern Pennsylvania,  Ohio,  Indiana  and  Illinois,  all  of  Michigan,  Wisconsin, 
Minnesota  and  the  Dakotas,  much  of  Iowa,  and  northeastern  Montana. 
The  glacier  had  a  southern  movement,  but  this  advance  southward  was, 
on  the  whole,  neutralized  by  the  melting  of  the  ice  on  the  southern  bor- 
der. In  cold  seasons,  the  movement  of  the  glacier  gained  on  the  power  of 
the  sun's  heat  to  melt  it,  and  it  advanced  southward.  In  warm  seasons, 
it  retreated  northward.  The  action  of  this  glacier  in  originating  and  modi- 
fying topographic  forms  was  twofold.  It  eroded  and  earned  away  material 
and  it  deposited  material.     It  is  the  latter  result  that  is  considered  here. 

The  material,  consisting  of  bowlders,  gravel,  and  sand  borne  by  the 
glacier  was  deposited  as  it  melted,  and  consequently  is  most  abundantly 
disti-ibuted  in  the  neighborhood  of  its  southern  boundary.  Owing  to  the 
recent  character  of  the  deposits,  they  have  been  little  eroded.  Lakes, 
swamps  and  waterfalls  abound  in  the  region  in  question.  The  terminal 
moraines  which  mark  the  limits  of  the  glacier  consist  of  an  irregular  mass 
of  material,  tkrown  down  in  the  greatest  confusion,  with  crooked  winding 
streams  and  sink  holes.  There  is  no  symmetry  or  law  in  its  disposition, 
but  it  is  made  up  of  details,  which  bear  no  relation  to  its  whole.  On  this 
account  it  must  be  sketched  piecemeal.  The  topographer  must  go  all  over 
it,  picking  up  each  detail  by  itself,  and  necessarily  the  control  must  be 
equally  minute. 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII-  PL.  XVI, 


DRUMLINS,    WIS. 

Scale    esTkoQ 
ContotLT  Interval  20  feet 


U.  S    GEOLOGICAL  SURVEY 


MONOGRAPH  XXII-  PL.  XVII 


A    PART  OF   THE  TERMINAL  MORAINE  AND  PITTED  PLAIN, WIS. 


Scale      eSTIT^o 

CoTLtartrIn.tei*val  20  £ee"t 


OEIGm  OF  TOPOGEAPHIO  POEMS.  123 

Within  the  limits  of  this  terminal  moraine,  the  commonest  character- 
istic feature  of  glacial  deposition  is  the  drumlin,  an  oval  mound  of  drift,  of 
height  ranging  from  a  few  feet  up  to  several  hundred  feet,  and  from  one  to 
several  square  miles  in  area.  They  ai-e  extremely  regular  in  shape  and 
their  curves  are  round  and  smooth.  In  many  localities  they  are  so  abun- 
dant as  practically  to  cover  the  surface,  the  intervals  between  them  being 
level  and  often  marshy.  The  axes  of  these  drumlins  are  commonly  par- 
allel, giving  a  curiously  artificial  appearance  to  the  map.  In  country  other- 
wise level,  they  determine  the  course  of  the  streams,  forcing  them  to  wind 
around  their  curves.  PL  xvi  shows  a  portion  of  the  drumlin  area  of 
southern  Wisconsin,  and  PI.  xvii  a  part  of  the  terminal  moraine  of  the 
same  region.  Pitted  plains,  which  are  level  areas  dotted  with  little  pits, 
are  common  features  of  glacial  action.  Osars,  or  long  winding  ridges,  are 
not  uncommon,  while  numerous  other  forms,  such  as  kettles,  etc.,  are  fre- 
quently seen,  but  are  of  less  importance  as  topographic  features. 

Glaciers  still  exist  in  the  Rocky  mountains,  the  Sierra  Nevada,  and  the 
Cascade  range,  though  they  are  by  no  means  as  extensive  as  in  former 
times.  At  the  bases  of  many  of  the  ranges  of  this  region  are  found  lateral 
moraines  reaching  out  from  the  mouths  of  mountain  gorges  and  connected 
at  their  ends  by  terminal  moraines. 

The  lateral  moraines  are  of  regular  form,  stretching  in  narrow  ridges, 
in  some  cases  parallel,  in  others  curving  away  from  one  another  from  the  foot 
of  the  canyon.  The  terminal  moraines  are  like  that  of  the  continental  glacier, 
confused  masses  of  material  heaped  up  in  disorder  and  consequently  diificult 
to  sketch  in  the  highest  degree. 

GLACIAL  EROSION. 

Glacial  erosion  is  very  similar  in  its  laws  and  action  to  aqueous  erosion, 
or  rather  to  that  part  of  it  which  is  called  corrasion.  The  principal  differ- 
ence between  them  lies  in  the  fact  that  ice  is  much  less  plastic  and  conse- 
quently does  not  accommodate  itself  so  readily  to  the  form  of  its  channel. 
It  moves,  too,  much  more  slowly  and  in  far  greater  volume  than  water. 

The  corrading  effect  of  the  continental  glacier  is  shown  in  northern 
New  England,  New  York,  Michigan,  Wisconsin,  and  Minnesota  very  mark- 


124  A  MANUAL  OF  TOPOGKAPHIC  METHODS. 

edly.  In  the  western  part  of  this  region  it  has  scoured  the  surface,  cutting 
av.-av  the  soft  rocks,  and  lea^^ng  the  hard  ones  in  projecting  knobs,  as  in  the 
^Marquette  Iron  range  of  Michigan.  This  work  was  done  so  recently  that 
the  drainage  systems  have  not  yet  been  well  developed.  The  streams  are 
tortuous  and  are  interrupted  by  lakes,  swamps,  and  rapids. 

In  northern  New  England  and  New  York  the  o-lacier  covered  a  regrion 
of  considerable  relief,  in  which  streams  had  established  deep  courses.  Much 
corrasion  was  done  by  it,  but  upon  its  retreat  the  streams  reoccupied  their 
former  beds. 

Most  of  the  gorges  of  the  Rocky  mountains  and  Sierra  Nevada,  which 
had  previously  ]:>een  excavated  by  streams,  have  been  occupied  by  glaciers, 
and  here  and  there  small  glaciers  may  still  be  found  at  their  heads.  These 
glaciers,  when  the}-  were  in  their  prime,  occupied  the  gorges  from  side  to 
side,  and  by  their  erosion  broadened  them  from  the  sharp  almost  V  shape 
which  water  corrasion  had  given  them  to  a  ^_^  shape,  similar  to  that  of  the 
bed  of  a  stream,  but  manv  times  larger. 

At  the  heads  of  the  main  gorge  and  many  of  its  branches,  where  tlie 
neve  fields  formerly  iniited  and  were  crowded  together  into  a  glacier  at  the 
heads  of  the  gorges,  there  is  commonly  an  amphitheater  with  steep,  even 
precipitous,  walls,  curving  around  in  a  semicircle.  In  the  middle  of  this  is 
sometimes  a  lake  or  pond,  with  a  rim  of  rock  inclosing  it  on  the  lower  side. 
This  lake  basin  was  scooped  out  by  the  glacial  ice,  as  it  came  together 
down  the  steep  slopes  of  the  amphitheater.  Here  the  ice  has  only  modified 
and  shaped  a  gorge  originally  carved  by  water.  Where  the  little  streams, 
flowing  toward  one  another  down  the  steep  mountain  side,  had  cut  many 
Kttle  gorges,  with  sharp  spurs  between  them,  the  ice  has  pared  away  the 
spm'S,  producing  an  amphitheater.  PL  xviii  illustrates  the  cirque  in  the 
Rocky  mountains  of  Colorado. 

DEPOSITION   FROM    THE  ATMOSPHERE. 

The  winds  transport  sand  and  deposit  it  in  di'ifts,  known  as  dunes, 
They  commonly  appear  as  lines  of  hills,  like  hogbacks,  with  the  gentle 
slope  toward  the  prevailing  winds.  Not  having  been  shaped  by  erosion, 
they  present  great  inequalities  of  surface. 


U.  S.  GEOLOGICAL  SURVEY. 


MONOGRAPH  XXII.  PL.  XVIII. 


A  PORTION  OF  THE  ELK  MTS.,  COL.,  SHOWING  AMPHITHEATRES. 


Scale    GsSoB 
Contour  IntervBl  100  feet 


EBPORTS. 


125 


SCALE  OF  FIELD   WORK. 

The  scale  iipou  which  the  field  ^vork  is  executed  is  commonly  larger 
than  that  upon  which  the  maps  are  to  be  published.  In  the  northeastern 
states  it  is  set  at  1:45000,  the  scale  of  publication  being  1:62500.  In 
the  southeastern  States  it  is  approximately  1  mile  to  an  inch,  the  scale  of 
publication  being  for  most  sheets  1:125000,  though  certain  sheets  in  Mary- 
land and  Florida  hdve  been  published  on  the  scale  1 :  62500.  In  the  Missis- 
sippi valley  it  is  uniformly  about  double  that  of  publication.  Where  the 
scale  of  publication  is  1 :  62500,  the  scale  of  field  work  is  2  inches  to  1  mile, 
and  where  the  former  is  1:125000,  the  latter  is  1  mile  to  an  inch.  In  the 
western  states,  the  scale  of  publication  being  1 :  125000,  the  field  sheets  are 
made  uniformly  on  the  scale  of  1  mile  to  an  inch. 

REPORTS. 

Each  field  party  is  required  to  make  a  monthly  report  to  the  chief  of 
division  and  the  chief  topographer  upon  the  progress  of  the  work  in  his 
party  during  the  month.  In  the  case  of  topograpliic  parties  these  reports 
are  made  upon  printed  forms,  of  which  the  following  is  a  sample : 

MONTHLY  REPORT  OF  TOPOGRAPHIC  PARTY. 

[To  be  made  out  in  duplicate  promptly  at  the  close  of  each  mouth,  one  copy  to  he  sent  to  the  geographer 
in  charge  of  the  division  and  one  copy  to  the  chief  topograi)her.] 

Department  of  the  Interior,  U.  S.  Geological  Survey, 


189 


Sir:  The  following  report  for  the  mouth  of 
topographic  party  under  my  charge : 
Names  and  positions  of  all  members  of  party,  - 
Instruments  used, 


,  189     ,  includes  a  statement  of  progress  of  the 


Barnard. 

Miller. 

Beall. 

Arrick. 

Triangulation  stations  occupied 

Points  located  by  triangulation 

Points  intersected  from  traverse 

Expended — for  salaries, 
Yours  respectfully, 


- ;  all  other  expenses,  $- 


- ;  total,  $- 


126 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Sheet.     Shade  surveyed  area. 

Upon  the  back  of  this  form  is  a  diagram  representing  an  atlas  sheet, 
as  above,  upon  which  is  to  be  indicated  the  area  surveyed  during  the  month. 

As  will  be  seen,  this  report  calls  for  statistics  concerning  the  control  of 
work,  specifying  secondary  triangulation,  traverse  and  the  measurements  of 
height,  together  with  the  areas  sketched. 


INSPECTION.  127  1 

INSPECTION. 

Inspection  of  the  work  is  done  by  the  chiefs  of  parties  and  of  divisions,  i 

and,  in  special  cases,  by  persons  detailed  by  them  for  this  purpose.  j 

The  accuracy  and  adequacy  of  the  control  are  shown  by  the  monthly  J 

reports  and  the  field  sheets   are  undergoing  constant  examination  from  the  I 

chiefs  of  party  and  of  division.     The  quality  of  the  sketching  is  examined  i 

on  the  ground.     As  far  as  possible  this  is  done  during  the  progress  of  the  j 

work,  using  the  field  sheets  as  soon  as  completed.     When  this  is  impracti-  ' 
cable,  it  is  done  during  the  succeeding  field  season,  using  photographs  of  the 
original  maps. 


CHAPTER   VI. 

OFFICE  WORK. 

The  office  work  of  the  topograpliers  consists  in  the  reduction  and  trans- 
fer of  the  work  from  field  sheets  to  the  original  maps.  The  reduction  is 
universally  effected  by  photography,  this  method  having  been  found  the 
most  accm'ate  and  economical  way  of  effecting  it. 

The  original  sheets  are  to  serve  as  the  original  record  of  work  and  as 
manuscript  for  the  engraver.  To  answer  these  purposes,  they  are  made 
complete  in  all  respects  as  to  hydrography,  hypsography,  and  public  cul- 
ture. Every  original  sheet  contains  within  itself  all  matter  which  is  to  be 
engraved  or  placed  on  record,  except  as  hereafter  noted. 

While  it  is  entirely  unnecessary  that  these  sheets  be  fine  specimens  of 
the  draftman's  skill,  they  are  workmanlike  in  appearance,  clear,  and  legible. 

The  original  sheets  are  commonly  drawn  upon  the  scale  upon  which 
they  are  to  be  published,  in  order  that  the  engraving  may  be  done  directly 
from  the  original  maps  rather  than  from  photographs  of  them.  Frequent 
departures  are,  however,  made  from  this  rule,  to  meet  other  requirements. 

The  contour  intervals  differ  widely  in  different  parts  of  the  country, 
ranging  from  6  feet  up  to  100  feet.  Where  the  scale  is  1 :  62500  the  cona- 
monest  contour  interval  is  20  feet.  In  Florida  and  Illinois  the  contour 
interval  is  reduced  to  10  feet,  while  in  the  low  alluvial  regions  of  southern 
Louisiana  it  is  only  5  feet. 

With  a  scale  of  1 :  125000  the  contour  interval  in  the  Apjjalachian 
mountain  region  is  100  feet,  in  the  Piedmont  region  it  is  50  feet,  and  upon 
the  Atlantic  plain  20  feet,  while  in  the  Dismal  swamp  of  Virginia  and  North 
Carolina  it  has  been  set  at  5  feet.  With  the  same  scale  in  Missouri,  Arkan- 
sas, and  eastern  Kansas  the  contour  interval  is  50  feet,  while  in  western 
Kansas  in  more  recent  work  it  is  20  feet.  In  Texas  the  coni  ,ar  interval 
128 


PEOJBCTIONS.  129 

rano-es  from  20  to  50  feet,  the  later  work  having  the  smaller  contour  inter- 
val. In  the  country  west  of  the  one  hundredth  meridian  the  contour 
interval  is  frequently  changed  with  the  alternation  of  mountain  and  valley, 
and  intervals  of  25,  50,  and  100  feet  are  employed,  the  interval  frequently 
changing  upon  the  same  sheet.  East  of  the  one  hundredth  meridian  the 
same  necessity  for  making  frequent  changes  in  contour  interval  does  not 
exist,  and  in  the  work  throughout  that  region  the  contour  interval  is  mii- 
form  upon  each  sheet. 

The  projection  used  is  the  polyconic,  each  sheet  being  projected  sepa- 
rately. 

Upon  Qriginals  to  be  pubhshed  upon  a  scale  of  1 :  62500  the  projection 
interval  is  5  minutes,  while  single  minute  lines  may  be  drawn  if  desired. 

The  construction  of  a  projection  upon  a  scale  of  1 :  62500  for  a  limited 
area  is  a  simple  matter,  but  requires  care  and  accuracy  and  the  use  of  the 
best  di-afting  instruments.  The  process  will  be  described  for  this  scale,  for 
which,  as  well  as  all  other  scales  heretofore  in  use,  tables  are  appended  to 
this  volume. 

First  draw  a  line  down  the  middle  of  the  sheet.  Lay  off  on  this  line 
the  length  of  the  several  projection  spaces  in  latitude.  Take  from  the  pro- 
jection table  for  the  scale  1:62500  the  length  of  5  minutes  of  latitude  and 
lay  it  off  repeatedly,  thus  establishing  the  points  of  intersection  of  parallels 
at  5  minutes  with  the  middle  meridian.  Through  these  points  draw  lines 
across  the  sheet  at  right  angles  to  the  middle  meridian,  using  beam  com- 
passes for  this  purpose.  Lay  off  on  these  hues  the  dm's  for  2'  30"  and  7' 
30"  from  the  middle  meridian,  con-esponding  to  the  latitude  on  each  side, 
and  at  these  points  erect  short  perpendiculars.  On  these  lay  off  the  dp's 
corresponding  to  the  dm's  and  through  the  points  thus  obtained  draw  and 
ink  the  projection  lines. 

For  other  scales  and  areas  the  process  is  quite  similar,  but  when  a 
large  area  such  as  that  of  the  United  States  is  to  be  projected,  the  mechan- 
ical difficulties  greatly  increase. 

Original  sheets  must  conform  in  size  and  shape  to  equal  parts  of  square 
degrees— i.  e.,  each  sheet  should  comprise  15'  of  latitude  by  15'  of  longitude, 
or  30'  in  each  dimension,  according  to  the  scale. 
MON  xxn 9 


130  A  MANUAL  OF  TOPOGRAPHIC  METHODS. 

COLORS    AND    CONVENTIONS. 

The  work  upon  the  original  sheets  conforms  to  the  system  of  conven- 
tions and  lettering  adopted  by  the  Survey.  Streams  must  be  inked  in  heavy 
Prussian  blue,  lettering  and  culture  in  India  ink,  and  contours  in  burnt  sienna. 
Indelible  inks  must  not  be  used  on  original  sheets.  Every  fourth,  or  fifth 
contoin-,  whatever  the  contour  interval,  should  be  empliasized,  in  order  to 
distinguish  it  from  the  others,  and  the  contours  so  distinguished  should  be 
freely  marked  in  columns  with  the  number  of  feet  above  sea  level  which 
they  indicate. 

Upon  the  map  should  be  located  all  towns  of  sufficient  importance  to 
contain  post-offices ;  all  railway  stations  and  other  settlements  of  any  impor- 
tance ;  all  houses,  all  public  roads,  and,  in  unsettled  regions,  the  principal 
trails;  all  railroads,  canals,  and  acequias;  all  tunnels  of  sufficient  length  to 
be  represented ;  bridges,  femes,  fords,  and  dams  upon  streams  of  sufficient 
importance  to  be  double-lined;  all  glaciers,  marshes,  sand,  and  sand  dunes, 
and  all  state,  county,  and  township  lines. 

The  convention  for  cities  and  towns  must  conform  as  closely  as  possible, 
in  extent,  du-ection  of  streets,  etc.,  to  the  actual  plan  of  the  place,  and  the 
houses  in  the  built  portion  should  be  blocked  in. 

Depression  contoiu-s  should,  if  they  inclose  large  areas,  be  indicated  by 
numbering  them  freely.  If  the  area  is  small,  they  should  be  hatched,  the 
hatchings  being  on  the  side  of  the  line  toward  the  depression. 

The  extent  of  forests  and  of  flood  plains  will  not  be  placed  upon  the 
original  maps,  but  should  be  colored  upon  photographs  of  them. 

TITLES   AND    LEGENDS. 

The  sheets  are  without  border  or  neat  line,  the  outer  projection  lines 
taking  the  place  of  the  latter.  Upon  the  margins  the  latitudes  and  longi- 
tudes of  the  projection  lines  must  be  given.  The  titles  and  legends  must 
conform  in  arrangement  and  character  to  those  on  the  printed  sheets. 

Wherever  it  is  practicable  to  do  so,  care  must  be  taken  to  connect  the  con- 
tours, streams,  and  culture  on  the  margins  of  sheets  with  the  adjoining  sheets. 

All  field  work  should,  if  possible,  be  platted  and  the  work  completed  during 
the  office  season  immediately  succeeding  the  field  work,  and  no  sheet  should 
be  reported  as  completed  until  it  is  ready  in  all  respects  to  be  engraved. 


ORIGINAL      SHEET. 


lOV/A 

WHEATLAND      SHEET 


Contour  Interval  20  fee 


APPENDIX. 


TABLES  FOR  COMPUTING  THE   DIFFERENCE   IN   THE   HEIGHT  OF  TWO  PLACES   FROM 
BAROMETRICAL  OBSERVATIONS. 

Table.  I. — J)  =  G015S.5Bx  log  H  or  h.     Argument:  The  observed  height  of  the  barometer  at  either  station. 

[Extracted  from  Smithsonian  Miscellaneous  Contributions.] 


Barom- 

Hundredtlis of  an  inch. 

Thou- 

Barom- 

eter ifl 

sandths 

eter  in 

Eng. 

of 

Eng. 

.OO 

.01 

.OS 

.03 

.04 

.OS 

.06 

.07 

.OS 

.09 

an 

inch. 

inc^. 

Eng.  ft. 

Sng.ft. 

Eng.ft. 

Ung.fl. 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Eng.  ft. 

Eng.ft. 

Feet. 

12.0 

4763.  4 

4785.  2 

4806.9 

4828.  7 

4850.4 

4872.1 

4893.  7 

4916.4 

4937.  0 

4938.  6 

12.0 

12.1 

4980.  2 

5001.8 

5023.  4 

5044.9 

5066. 4 

5087.  9 

5109.4 

5130.9 

5152.  4 

5173.  8 

12.1 

12.2 

5195.  2 

5216.  6 

5238.  0 

5259. 4 

5280.  7 

5302. 1 

5323.  4 

5344.  7 

5367.  0 

5387.  2 

12.2 

12.3 

5408.  5 

5429.  8 

5432.  0 

5472.  2 

5493.4 

5514.  5 

5535.7 

5556.  8 

5578. 9 

5599.  0 

1 

2.1 

12.3 

12.4 

5620. 1 

5641.  2 

5662.  2 

5683. 2 

5704. 3 

5725.  3 

5746.  2 

5767.  2 

5788. 1 

5809.  0 

2 

4.2 

12.4 

12.5 

5829.  9 

5850.8 

5871.  7 

5892.  6 

5913. 4 

5931.2 

5955.  0 

5975.  8 

5996.  6 

6017.  4 

3 

6.2 

13.5 

12.6 

6038. 1 

6058. 8 

6079.  6 

6100.  2 

6120.  9 

6141.6 

6162.  2 

6182.  8 

6203.  5 

6234.  0 

4 

8.3 

12.6 

12.7 

6244.6 

6265.  2 

6285.  8 

6306.  3 

6326.  8 

6347.  3 

6367.  8 

6388. 3 

6408.  8 

6429.  2 

5 

10.4 

12.7 

12.8 

6449.  6 

6470.  0 

6490. 4 

6510.  8 

6531. 1 

6551. 5 

6571.  8 

6592. 1 

6612.  4 

6632.  7 

6 

12.5 

12.8 

12.9 

6652.  9 

6673. 2 

6693.  4 

6713.  6 

6733.  8 

6754. 0 

6774. 1 

6794.  3 

6814.  4 

6834.  5 

7 

14.6 

12.9 

13.0 

6854.  7 

6874.  7 

6894.  8 

6914.  9 

6934.  9 

6955.0 

6975.  0 

6995.  0 

7014.  9 

7034.9 

8 

16.6 

13.0 

13.1 

7054. 9 

7074.  8 

7094.  7 

7114.  6 

7134.  5 

7154.4 

7174.  3 

7194.  I 

7213.  9 

7233.  8 

9 

18.7 

13.1 

13.2 

7253.  6 

7273.  3 

7293. 1 

7312.  9 

7332.  6 

7352.  3 

7372. 1 

7391.  8 

7411.  4 

7431. 1 

13.2 

13.3 

7450.8 

7470.  4 

7490.  0 

7509. 6 

7529.  2 

7548.  8 

7568.  4 

7587. 9 

7607.  4 

7627.  0 

13.3 

13.4 

7646.  5 

7666.  0 

7685.4 

7704.  9 

7724.4 

7743. 8 

7763.  2 

7782.  6 

7802.0 

7821.  4 

13.4 

13.5 

7840.  8 

7860. 1 

7879.4 

7898.  7 

7918.  0 

7937. 3 

7956.  6 

7975.  8 

7995. 1 

8014. 3 

13.5 

13.6 

8033.  6 

8052.8 

8071.9 

8091. 1 

8110.3 

8129.4 

8148. 6 

8167.7 

8)86.8 

8205. 9 

13.6 

13.7 

8225.  0 

8244.0 

8263. 1 

8282. 1 

8301.1 

8320. 1 

8339. 1 

8358. 1 

8377. 1 

8396.  0 

1 

1.9 

13.7 

13.8 

8415.  0 

8433. 9 

8452.  8 

8471.  7 

8490.  6 

8509.  4 

8528.  3 

S547. 1 

8565.  9 

8574.8 

2 

3.8 

13.8 

13.9 

8603.  6 

8622.  3 

8641. 1 

8659.  9 

8678.  6 

8697.4 

8716. 1 

8734.  8 

8753.  5 

8772.  2 

3 

5.6 

13.9 

14.0 

8790.  8 

8809. 5 

8828.  2 

8846.  8 

8865.4 

8884.  0 

8902. 6 

8921.2 

8939.  7 

8958.  3 

4 

7.5 

14.0 

14.1 

8976.  8 

8995.  4 

9013.9 

9032.  4 

9050. 8 

9069. 3 

9087.  8 

9106.  2 

9124. 6 

9143. 0 

5 

9.4 

14.1 

14.2 

9161.  4 

9179.  8 

9198.  2 

9216.  6 

9234  9 

9253.  3 

9271.  6 

9289.  9 

9308.  2 

9326.  5 

6 

11.3 

14.2 

14.3 

9344. 7 

9363. 0 

9381.  3 

9399. 5 

9417.  7 

9436.  0 

9454.  2 

9472. 3 

9490.  5 

9508. 7 

7 

13.2 

14.3 

14.4 

9526. 8 

9545.0 

9563. 1 

9581.  2 

9599.  3 

9617. 4 

9635.  5 

9653.  5 

9671.6 

9689. 6 

8 

15.0 

14.4 

14.5 

9707.  6 

9725.  7 

9743,7 

9761.  7 

9779.  6 

9797.  6 

9815.  6 

9833.  5 

9831.4 

9869.  3 

9 

17.0 

14.5 

14.6 

98S7.  2 

9905. 1 

9923. 0 

9940.  9 

9958.  7 

9976.  5 

9994.4 

10012. 2 

10030. 0 

10047.  8 

14.6 

14.7 

10065.  5 

10083.  3 

10101.1 

10118.  8 

10136. 6 

10154.  3 

10172.  0 

10189,  7 

10207.  4 

10225. 1 

14.7 

14.8 

10242. 7 

10260.  4 

10278. 0 

10295.  7 

10313.  3 

10330.  9 

10348.  5 

10366.  1 

10383.  6 

10401. 2 

1 

1.7 

14.8 

14.9 

10418.  7 

10436. 3 

10453.8 

10471.  3 

10488. 8 

10506. 3 

10523.  7 

10541. 2 

10558. 6 

10576.  0 

2 

3.4 

14.9 

15.0 

10593. 4 

10610.  8 

10628.  2 

10645.  6 

10662.  9 

10680.  3 

10697.  6 

10715.  0 

10732.  3 

10749.  6 

3 

5.1 

15.0 

15.1 

10766.  9 

10784. 1 

10801. 5 

10818.7 

10836.0 

10853.  2 

10870.  5 

10887.  7 

10904.  9 

10922. 1 

4 

6.8 

13.1 

15.2 

10939.  3 

10956.  5 

10973.  6 

10990.  8 

11008.  0 

11025. 1 

11042.  2 

11059.  3 

11076.  4 

11093.  5 

5 

8.5 

15.2 

15.3 

11110.  6 

11127.7 

11144. 7 

1116L8 

11178.8 

11195.  8 

11212.  8 

11229.  8 

11246.  8 

11263. 8 

6 

10.2 

15.3 

15.4 

11280.  8 

11297.  8 

11314.  7 

11331.6 

11348.  6 

11365.  5 

11382.  4 

11399.  3 

11416.2 

11433.0 

7 

11.9 

13.4 

15.5 

11449.  9 

11466.  7 

11483.  6 

11500.  4 

11517.  2 

11534. 0 

11550.8 

11567.  6 

11584.  4 

11601.1 

8 

13.6 

15.3 

15.6 

11617.9 

11634.  6 

11651.4 

11668. 1 

11684.8 

11701.  5 

11718.2 

11734. 9 

11751.  6 

11768: 2 

9 

15.3 

15.6 

15.7 

11784.  9 

11801.  5 

11818.  2 

11834.  8 

11851.4 

11868.  0 

11884.  6 

11901.1 

11917. 7 

11934.  3 

15.7 

15.8 

11950.  8 

11967.  3 

11983.8 

12000. 4 

12016.  9 

12033. 3 

12049.  8 

12066.  3 

12082.  7 

12099.  2 

15.8 

15.9 

12115.  6 

12132.0 

12148.  4 

12164.  8 

12181.  2 

12197.6 

12214.  0 

12230.  4 

12246. 7 

12263. 1 

13.9 

16.0 

12279.  6 

12295.  9 

12312. 2 

12328.  5 

12344.  8 

12361. 1 

12377.4 

12393.  6 

12409.  9 

12426. 1 

16.0 

16.1 

12442.  4 

12458.  6 

12474.  8 

12491.  0 

12507.  2 

12523.4 

12539.  6 

12555.  7 

12571. 9 

12588.  0 

16.1 

16.2 

12604.  2 

12620.  3 

12636.  4 

12652.  5 

12668.  6 

12684. 7 

12700.  8 

12716.  8 

12732.  9 

12748.  9 

1 

1.6 

16.2 

16.3 

12765.  0 

12781.  0 

12797.  0 

12813.  0 

12829.  0 

12845.0 

12861.  0 

12876.  9 

12893.  9 

12908.8 

2 

3.1 

16.3 

16.4 

12924.  8 

12940.  7 

12956. 6 

12972.  5 

12988.4 

13004.  3 

13020.  2 

13036.  0 

13051.  9 

13067.  7 

3 

4.7 

16.4 

16.5 

13083.6 

13099.4 

13115.  2 

13131.0 

13146.  8 

13162.  6 

13178.4 

13194.  2 

13210.  0 

13225.  7 

4 

6.3 

16.5 

16.6 

13241.5 

13257.  2 

13272.  9 

13288.  6 

13304. 3 

13320.  0 

13335. 7 

13351.5 

13367. 1 

13382.  7 

5 

7.8 

16.7 

16.7 

1.3398. 4 

13414.  0 

13429.  6 

1344.5.3 

13460.  8 

13476.4 

13492.  0 

13507.6 

13523. 2 

13538.  7 

6 

9.4 

16.7 

16.8 

13554.  3 

13569.  8 

13585.  4 

13600.  9 

13616.4 

13631. 9 

13647.  4 

13662. 9 

13678.  4 

13693.  9 

7 

11.0 

16.8 

16.9 

13709. 4 

13724. 8 

13740.  3 

13755.7 

13771. 1 

13786.  5 

13801.9 

13817.  3 

13832.  7 

13848. 1 

8 

12.5 

16.9 

17.0 

13863.  5 

13878.  8 

13894.  2 

13909.6 

13924.  9 

13940.  2 

13955.  6 

13970. 9 

13986.  2 

140O1.  5 

9 

14.1 

17.0 

17.1 

14016.  8 

14032.  0 

14047. 3 

14062.  6 

14077.  8 

14093.  0 

14108.  3 

14123.  f) 

14138.  7 

14153.  9 

17.1 

17.2 

14169. 1 

14184.3 

14199.4 

14214.  6 

14229.8 

14244.  9 

14260. 1 

14275.  2 

14290. 3 

14305.  5 

17.2 

17.3 

14320.  6 

14335.  7 

14350.8 

14365.8 

14380. 9 

14396.  0 

14411.0 

14426. 1 

14441. 1 

14456.  2 

17.3 

17.4 

14471.2 

14486.2 

14501.2 

14516.2 

14531.2 

14546. 1 

14561. 1 

14576. 1 

14591.0 

14605.  9 

1 

1.5 

17.4 

131 


132 


A  MANUAL  OF  TOPOGEAPHIG  METHODS. 


Table. 

I.— D= 

60158.58  ?o<?xH 

0,h.      A 

rgiiment 

:  TheoJ} 

served  height  of  b 

urometer  at  eithei 

station. — 

Cont'd. 

Barom- 

Hundredths of  an  inch. 

Thou- 

Barom- 

eter in 
incE. 

sandths 

of 
an  inch. 

eter  in 
a. 

.OO 

.Ol 

.03 

.03 

.04 

.05 

.06 

.07 

.08 

.09 

Eng.ft. 

Eng.ft. 

JEng.  ft. 

Eng.ft. 

Enq.ft. 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Feet. 

17.5 

14620.  9 

14635.  8 

14C.-.U.  7 

14064.  6 

14080.  5 

14695. 4 

14710. 3 

14725.  2 

14740. 1 

14754.  9 

2 

2.9 

17.5 

17.6 

14769.  8 

14784.  6 

1479!).  4 

11S14.:! 

14829.  1 

14843. 9 

14858. 7 

14873.  5 

14888.  2 

14903.  0 

3 

4.4 

17.6 

17.7 

14917.  8 

14932.  5 

14947. :! 

149lii;.  0 

14976.  8 

14991.5 

15000.  2 

1.5020.  9 

13035.  6 

15050,  3 

4 

5.8 

17.7- 

17.8 

15065.  0 

15079. 6 

151194.  3 

Ifiliiu.  0 

15123.6 

15138.  2 

15152.  9 

16107. 5 

15122. 1 

1.5196.7 

5 

7,3 

17.8 

17.9 

15211. 3 

15225.  9 

15240.  5 

15255.  0 

15269.  6 

15284.  2 

15298.  7 

15313. 3 

15327.  8 

15342.  4 

6 

8.8 

17.9 

18.0 

15356.  8 

15371. 3 

15385.  8 

15400.  3 

15414.  8 

15429. 3 

15443.7 

15458. 2 

15472.  7 

15487.1 

7 

10.2 

18.0 

18.1 

15501.  5 

15516.  0 

15530.  4 

15544.  8 

15559.  2 

15573. 6 

15588,  0 

15602.  4 

15616. 8 

15631.  2 

8 

11.7 

18.1 

13.2 

15645.  5 

15659.9 

15674. 2 

15688.  5 

15702.  9 

15717. 2 

15731. 5 

15745.  8 

15760. 1 

15774.4 

9 

13.1 

18.2 

18.3 

15788.  6 

15802.  9 

15817,  2 

15831.4 

15845.  7 

15359.  9 

15874,  2 

15888.  4 

15902.  6 

15916.  8 

18.3 

18.4 

15931.  0 

15945.2 

15959.  4 

15973.  6 

15987.  8 

16001. 9 

16016. 1 

16030.2 

16044.4 

16058.  5 

18.4 

18.5 

16072.  6 

16086.  8 

16100.  9 

16115.0 

16129, 1 

16143.  2 

16157.  3 

16171,3 

16135.  4 

16199.  5 

18.5 

18.6 

16213.  5 

16227.  6 

16241.  6 

162.55.6 

16269.  7 

16283.  7 

16297.  7 

16311,7 

16325.7 

16339.  6 

18.6 

18.7 

16353.  5 

16367.  5 

16381.  5 

16395.  4 

16409.  4 

16423.  3 

16437.  2 

16451.2 

16465. 1 

16479.  0 

1 

1.4 

18.7 

18.8 

lb492.  9 

16506.  8 

16520. 7 

16534.  6 

16548.  5 

16562.  3 

16576.  2 

16590.  0 

16603. 9 

16617.  8 

2 

2.7 

18.8 

18.9 

16631.  5 

16645.  4 

16659.  2 

16673.  0 

16686.  8 

16700. 6 

16714.4 

16728. 1 

16741. 9 

16755.7 

3 

4.1 

13.9 

19.0 

16769. 4 

16783.  2 

16796.  9 

16810.  6 

16824.  3 

16838. 1 

16851.8 

16865.  5 

16879. 2 

16892.8 

4 

5.4 

19.0 

19.1 

16906.  5 

16920.  2 

16933.  9 

16947.  5 

16961.  2 

16974.  9 

16988.  5 

17002. 1 

17015.  8 

17029.4 

5 

6.3 

19.1 

19.2 

17043. 0 

17056.  6 

17070. 2 

17083.  S 

17097.  4 

17110.9 

17124.5 

17138. 1 

17151.6 

J7165.2 

6 

8.1 

19.2 

19.3 

17178.  7 

17192.  2 

17205.  8 

17219.  3 

17232.  8 

17246.  3 

17259.  8 

17273.3 

17286.  8 

17300.  3 

7 

9.5 

19.3 

19.4 

17313.7 

17327.  2 

17340.  6 

17354. 1 

17367.  5 

17380.  9 

17394.  4 

17407.  8 

17421.  2 

17434.6 

"  8 

10.9 

19.4 

19.5 

17448. 0 

17461.  4 

17474. 8 

17488.  2 

17501.6 

17515.  0 

17523.  3 

17541.  7 

17555.  0 

17568.4 

9 

12.2 

19.5 

19.6 

17581.  7 

17595.  0 

17608.  3 

17621.  7 

17635.  0 

17648.  2 

17661. 5 

17674.  8 

17688. 1 

17701.4 

19.6 

19.7 

17714.  6 

17727.  9 

17741.1 

17754.  4 

17767.  6 

17780.8 

17794. 1 

17807.3 

17820.  5 

17833.  7 

19.7 

19.8 

17846.  9 

17860. 1 

17873.  3 

17886. 5 

17899.  6 

17912.  8 

17926. 0 

17939. 1 

17952.  2 

17965.4 

19.8 

19.9 

17978.  5 

17991.  6 

18004.  8 

18017.  9 

18031.  0 

18044. 1 

18057,  2 

18070.  3 

13083.  4 

13096.4 

1 

1.3 

19.9 

20.0 

18109.5 

18122.  6 

18135.6 

18148.7 

18161.7 

18174.  8 

18187,  8 

18200. 3 

13213.  8 

13226.  8 

2 

2.6 

20.0 

20.1 

18239.  8 

18252.  8 

18265.  8 

18278.  8 

18291.  S 

13304. 8 

18317.7 

18330.7 

18343.  6 

18356.6 

3 

3.9 

20.1 

20.2 

18369.  5 

18382. 5 

18395.4 

18408.3 

18421.2 

18434. 1 

13447.  0 

18459.  9 

18472. 3 

18435.  7 

4 

5.1 

20.2 

20.3 

18498.  5 

18511.4 

18524.  3 

18537.1 

18550.  0 

18502.  8 

18575.  7 

13.588.  5 

18601.3 

18614. 1 

5 

6.4 

20.3 

20.4 

18626.  9 

18639.  7 

18652. 5 

18665.  3 

18678. 1 

18690.  9 

18703.  6 

18716.4 

18729. 1 

18741. 9 

6 

7.7 

20.4 

20.5 

18754.  6 

18767.  4 

18780. 1 

18792.  9 

18805.  6 

18818. 3 

18831.  0 

18843.  7 

13856.4 

13869. 1 

7 

9.0 

20.5 

20.6 

18881.  S 

18H94.  3 

18907.  2 

18919.9 

18932.5 

18945.  2 

18957.  8 

13970.  5 

18983. 1 

18995.  7 

8 

10.3 

20.6 

20.7 

190U8. 3 

19021.0 

19033. 6 

19046.  2 

19058.  8 

19071.  4 

19083.9 

19096.  5 

19109. 1 

19121.  7 

9 

11.6 

20.7 

20.8 

19134. 2 

19140.  8 

19159.  3 

19171.9 

19184.  4 

19196.  9 

19209. 5 

19222.  0 

19234.  5 

19247.  0 

20.8 

20.9 

19259. 5 

19272.  0 

19284.  5 

19297. 1 

19309.  5 

19322.  0 

19334.  4 

19346.  9 

19359.  3 

19371.  8 

20.9 

21.0 

19384.  3 

19396.  7 

19409. 1 

19421.5 

19434.0 

19446.  4 

19458:  8 

19471.  2 

19483.  6 

19496.  0 

1 

1.2 

21.0 

21.1 

19508.  4 

19520.  8 

19533. 1 

19545.  5 

19557.  9 

19570.  2 . 

19589.6   19594.9 

19607.  3 

19619.  6 

2 

2.4 

21.1 

21.2 

19632.  0 

19644.3 

19656.  6 

19668.  9 

19681.  2 

19693. 5 

19705.  8 

19718.  0 

19730.  3 

19742.  6 

3 

3.6 

21.2 

21.3 

19754.  9- 

19767. 1 

19779. 4 

19791.  6 

19803.9 

19816. 1 

19828.4 

19340.  6 

19852. 8 

19865,  0 

4 

4.8 

21.3 

21.4 

19877.  3 

19889.  5 

19901.  7 

19913.  9 

19926,  0 

19938.  2 

19950.  4 

19962.  6 

19974. 7 

19986.  9 

21.4 

21.5 

19999. 1 

20011.  2 

20023.  3 

20035.  5 

20047.  6 

20059.  7 

20071.  8 

20033.  9 

20096. 1 

20108.  2 

5 

6.0 

21.5 

21.6 

20120.  3 

a0132.  3 

20144. 4 

20156.  5 

20163.  6 

20180.  7 

20192.  7 

20204.  8 

20216.  9 

20228.  9 

6 

7.2 

21.6 

21.7 

20241.  0 

20253.  0 

20265  0 

20277.6 

20289. 1 

20301. 1 

20313. 1 

20325. 1 

20337. 1 

20349. 1 

7 

8.4 

21.7 

21.8 

20361. 1 

20373. 0 

20385. 0 

20397.0 

20409. 0 

20420. 9 

20432.  9 

20444.  8 

20456.  8 

20468. 7 

3 

9.7 

21.8 

21.9 

20480. 7 

20492.  6 

20504.  5 

20516.  4 

20523.  3 

20540.  2 

20552. 1 

2U564.  0 

20575.  9 

20587.  8 

9 

10.9 

21.9 

22.0 

20599. 7 

20611.  5 

20623.  4 

20635.  3 

20647. 1 

20659.  0 

20670.  8 

20682.  7 

20694.  5 

20706. 3 

22.0 

22.1 

20718.2 

20732.  0 

20741.  8 

20753.  6 

20765.4 

20777.  2 

20789.  0 

20801.  8 

20812.6 

20824.  4 

22.1 

22.2 

20836.  2 

20847. 9 

20859.  7 

20871.4 

20883.  2 

20894.  9 

20906.  7 

20918.  4 

20930. 1 

20941. 9 

22.2 

22.3 

20953.  6 

20965.  3 

20977.  0 

20988.  7 

21000. 4 

21012. 1 

21023.  8 

21035.  4 

21047. 1 

21058.  8 

1 

1.1 

22.3 

22.4 

21070. 5 

21082. 1 

21093. 8 

21105.  4 

21117. 1 

21128.7 

21140.4 

21152.  0 

21163.  6 

21175.  3 

2 

2.3 

22.4 

22.5 

21186.  9 

21198.  5 

21210. 1 

21221.  6 

21233.  2 

21244.  8 

21256. 4 

21268.  0 

21279.  5 

21291. 1 

3 

3.4 

22.5 

22.  G 

21302.  6 

21314.  2 

21325.  8 

21337.  3 

21348.  9 

21360.  4 

21371.  9 

21383.5 

21395.  0 

21406.  5 

4 

4.6 

22.6 

22.7 

21418. 1 

21429.  6 

21441. 1 

21452. 5 

21464.  0 

21465.5 

21487.0 

21498.  5 

21509.9 

21521.4 

5 

5.7 

22.7 

22.8 

21532. 9 

21544.  3 

21555.  8 

21567.  2 

21578.7 

21590. 1 

21601.  6 

21613.  0 

21624.4 

21635.  8 

6 

6.8 

22.8 

22.9 

21647.3 

21658.  7 

21670. 1 

216S1.4 

21692. 8 

21704.2 

21715.  6 

21727. 0 

21738.  3 

21749.  7 

7 

8.0 

22.9 

23.0 

21761.  0 

21772.4 

21783.  7 

2179.5. 1 

21806.  4 

21317.7 

21829. 1 

21840.4 

21851.7 

21863.  0 

8 

9.1 

23.0 

23.1 

21874. 3 

21885.  6 

L'ls  '7  n 

:j!'.'(|.:,  :; 

21919.  6 

21930.  8 

21942. 1 

219.53.4 

21964.  7 

21976.0 

9 

10.2 

23.1 

2.S.2 

21987.2 

21998. 5 

_  .  0 

22032.  3 

22043. 5 

22054.  7 

22066,  0 

22077. 2 

22088. 4 

23.2 

23.3 

22099.  6 

22110. 8 

22144. 5 

22155.6 

22166.8 

22173,  0 

22189. 2 

22200.  4 

23.3 

23.4 

22211.  5 

22222.  7 

--:■;.' 

I'JJ'.'.-O 

22256.  2 

22267.  3 

22278.  4 

22289.  6 

22300.  7 

22311.  8 

23.4 

23.5 

22322. 9 

22334.  0 

22345.  2 

22356. 3 

22367. 4 

22378.  4 

22389.  5 

22400.  6 

22411.  7 

22422. 8 

23.5 

23.6 

22433.  8 

22444. 9 

22456. 0 

22467.  0 

22478. 1 

22439. 1 

22500.  2 

22511.  2 

22522.  3 

22533.  3 

23.6 

23.7 

22544.  3 

22555.  4 

22566.4 

22577.4 

22588.4 

22599.4 

22610.4 

22621.4 

22632.  4 

22643.4 

23.7 

23.8 

22654.3 

22665.3 

22676.  3 

22687.  2 

22698.2 

22709. 1 

22720.1  1  22731.0 

22742.0 

22752.  9 

1 

1.1 

23.8 

23.9 

22763.  8 

22774.  8 

22785  7 

22796  6 

22807  5 

22818  4 

22829  4 

22840.  3 

22851.  2 

22862.  0 

2 

2.2 

23.9 

24.0 

22873.  0 

22883.9 

22894  7 

2^90  ( 

116  5 

22927  4 

22939  2 

22949. 1 

22960.0 

22970.  8 

3 

3.2 

24.0 

24.1 

22981.  7 

229a2. 5 

23J0i  3 

14 

0 

lOJ  h 

104b  6 

23057. 5 

23068.  3 

23079. 1 

4 

4.3 

24,1 

34.2 

23089.  9 

2iiao. 7 

23111  4 

1   > 

14  8 

2  1d4  5 

23165.  3 

23176. 1 

23136,  3 

6 

5.4 

24.2 

24.3 

23197.  6 

23208.  3 

23219  1 

1   0 

ol  o 

21-62  0 

21272.7 

23283.  4 

23294, 2 

6 

6.5 

24.3 

21.4 

23304.  9 

23315.  6 

23J2fa 

" 

-  .,4  0 

2J35b  3 

2  u09  0 

23379.  7 

23390. 3 

23401.  0 

7 

7.5 

24.4 

BAROMETKIC  TABLES. 


133 


Table.  I. — 0=^60158.58  x  log  H  or  li.     Argtimeni:     The  ohiserved height  of  the  barometer  at  either  station- 

Continued. 


Barom- 

Hundredth 

3  of  an  inch. 

Thou- 

Barom- 

eter in 
Bng. 
iucE. 

sandths 

of 
an  inch. 

eter  in 
Eng. 

.00 

.01 

.02 

.03 

.04 

.05 

.06 

.07 

.OS 

.09 

Eng.ft. 

Eng.ft. 

Enij./t 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Eng.ft. 

Eng.  ft. 

Eng.ft. 

Eng.ft. 

Feet. 

24.5 

23411.7 

23422.  3 

23433.  0 

23443.  7 

23464.3 

23464.  9 

23475.  6 

23486.  2 

23496.  8 

23607.  4 

8 

8.6 

24.5 

24.  G 

23518. 1 

23528.  7 

23539. 3 

23549.  9 

23660.  5 

23571. 1 

23681. 7 

23592.  3 

23602. 9 

23613.  6 

9 

9.7 

24.6 

24.7 

23624. 1 

23634.  6 

23645.  2 

23655.  8 

23666.  3 

23676.  9 

23687.5 

23698.0 

33708. 6 

23719. 1 

24.7 

24.8 

23729.  7 

23740.  2 

33750. 7 

23761.2 

23771.7 

23782.  3 

33792.  8 

23803. 3 

23813.8 

33824.  3 

24.8 

24.9 

23834.  8 

23845.  3 

23855.  7 

23866. 2 

23876.7 

23887.  2 

33897.  7 

23908. 2 

23918.  6 

23929. 1 

1 

1.0 

34.9 

25.0 

23939.  5 

23949.  9 

23960.  4 

23970.  8 

23981.  3 

23991. 7 

24002. 1 

24012. 5 

34023.  0 

24033.4 

2 

2.1 

25.0 

25.1 

24043. 8 

24054. 2 

24064.  6 

24075.  0 

24085.  4 

24095.  7 

24106.1 

24116,5 

24126.  9 

34137.  2 

3 

3.1 

26.1 

25.2 

24147.  6 

24158.  0 

24168. 3 

24178.  7 

24189.  0 

24199.4 

24209. 7 

24220. 1 

34230.  4 

24240.  8 

4 

4.1 

25.2 

25.3 

24251. 1 

24261.  4 

24271. 8 

24282. 1 

24292.  4 

24303.  7 

24313. 0 

24323.  3 

24333.  6 

24343.  9 

5 

5.1 

25.3 

25.4 

24354.  2 

24364.  5 

24374.  7 

24385.  0 

24395.  3 

24406. 5 

24415.  8 

24426. 1 

24436. 3 

24446.  6 

6 

6.2 

26.4 

25.5 

24456.  8 

24467.  0 

24477.  3 

24487.  5 

24497. 8 

24508.  0 

24518.  2 

24528.  4 

24638.  7 

24548.  9 

7 

7.2 

26.5 

25.6 

24559. 1 

24569.  3 

24579.  5 

24589.7 

24599. 9 

24610.  0 

24630.  2 

24630.4 

24640.  6 

24650.7 

8 

8.2 

26.6 

25.7 

24660.  9 

24671. 1 

24681. 2 

24691.4 

24701.  6 

24711. 7 

34721.  8 

24732. 0 

•  24742, 1 

24752.  3 

9 

9.2 

26.7 

25.8 

24762.4 

24772.  5 

24782.  6 

24792.  8 

24802,  9 

24813.  0 

24833. 1 

24833.  2 

24843.  3 

24853.  4 

25.8 

25.9 

24863.  5 

24873. 6 

24883. 7 

24893.7 

24903. 8 

24913.  9 

24931. 0 

24934.0 

24944. 1 

24964. 1 

25.9 

26.0 

24964.  2 

24974.  2 

24984.  3 

24994.  3 

25004. 4 

25014. 4 

26024. 4 

26034.  4 

25044. 5 

25064.  5 

26.0 

26.1 

25064. 5 

25074.  5 

25084.  5 

25094.  5 

25104.  5 

25114.  6 

25124.  5 

25134.  5 

26144.  4 

25154.  4 

26.1 

26.2 

25164. 4 

25174.  4 

25184.  3 

25194.  3 

25204. 2 

25214.  2 

25224. 1 

25334. 1 

25244.  0 

25254.  0 

1 

1.0 

26.2 

26.3 

25263.  9 

25273. 8 

25283.  8 

25293.  7 

25303. 6 

25313. 6 

25323.  4 

35333.  3 

25343.  2 

2,5353. 1 

3 

2.0 

26.3 

26.4 

25363.  0 

25372.9 

25382.8 

25392.  7 

25402.  6 

25412. 4 

26422. 3 

25432.  2 

26442. 1 

25451.  9 

3 

2.9 

26.4 

26.5 

25461.  8 

2.';471.7 

25481.  5 

25491.4 

25501. 2 

25511.  0 

25620.  9 

25530.  7 

25540.  6 

25650. 4 

4 

3.9 

26.5 

26.0 

25560.  2 

25570.  0 

25579.8 

25589.  7 

25599. 5 

25609. 3 

25619. 1 

25628.  9 

25638. 7 

25643. 5 

5 

4.9 

26.6 

26.7 

25658. 3 

25668. 1 

25677.8 

25687.  6 

25697. 4 

26707. 1 

25716.  9 

25726.  7 

25736.4 

25746.  2 

6 

5.9 

26.7 

26.8 

25755.  9 

25765.  6 

25775.  4 

25785. 1 

25794.  8 

25804. 6 

25814.  3 

26824.  0 

25833.  3 

25843. 5 

7 

6.9 

26.8 

26.9 

25853.  2 

25862.  9 

25872.  6 

25882.  3 

25893. 0 

25901.  7 

25911.4 

25921, 1 

25930.  8 

25940.5 

8 

7.8 

26.9 

27.0 

25950.  2 

25959. 9 

25969.  6 

25979.  2 

25988.9 

25998.  6 

26008.  2 

26017.  9 

26027. 5 

26037. 2 

9 

8.8 

27.0 

27.1 

26046.  8 

261156.  5 

20066.  1 

26075. 7 

26085.  3 

26095.  0 

26104.  6 

26114.  2 

26123. 8 

26133.4 

27.1 

27.2 

26143. 0 

26152.6 

26162.  2 

26171.  8 

26181.  4 

26191.  0 

26200.  6 

26210.  2 

26219. 8 

26339.  3 

27.2 

27.3 

26238.  9 

26248.  0 

20258.  0 

26267.  6 

26277. 2 

26286.  7 

26296.  3 

26306. 8 

26315.  3 

36324.  9 

27.3 

27.4 

26334.  4 

26344.  0 

26353.  5 

26363.  0 

26372.  5 

26382.1 

20391.  6 

26401. 1 

26410.  6 

26420. 1 

1 

0.9 

27.4 

27.5 

26429.  6 

26439. 1 

26448.  6 

26458. 1 

26467.  6 

26477. 1 

26436.  5 

26496.  0 

26505. 5 

26514,  9 

2 

1.9 

27.5 

27.6 

26524.  4 

26533.  9 

26543.3 

26552.  8 

26562.  3 

26571.7 

26681.  2 

26590.  6 

26600.  0 

26609.  5 

3 

2.3 

27.6 

27.7 

26618.  9 

26628.  4 

26637.  8 

26647. 2 

26656.  7 

26066. 1 

26676. 5 

26684.  9 

26694.  3 

26703.  7 

4 

3.7 

27.7 

27.8 

26713. 1 

26722.  5 

26731.  9 

20741.  3 

26750.  7 

26760. 1 

26769.  6 

26778.  8 

26788.  2 

26797. 6 

5 

4.7 

27.8 

27.9 

26806.  9 

26816.  3 

26825.  6 

26835.  0 

26844.  3 

26853. 7 

20863.0 

26872.  3 

26881.  7 

36891.  0 

6 

5.6 

27.9 

28.0 

26900.  4 

26909.  7 

21919.  0 

26928. 4 

26937. 7 

26947.  0 

26956.  3 

26965.  6 

26975.  0' 

36984.  3 

7 

.  6.5 

28.0 

28.1 

26993.  6 

27002. 9 

27012.  2 

27021.5 

27030. 7 

27040.  0 

37049,  3 

27058.  6 

27067. 8 

27077. 1 

8 

7.5 

28.1 

28.2 

27086.  4 

27095.  6 

27104.  9 

27114. 3 

27123.  4 

27132.  7 

37141.  9 

27151.2 

27160.  4 

27169,  6 

9 

8.4 

28.2 

28.3 

27178. 9 

27188. 1 

27197.  3 

27206.  6 

27215.  7 

27225.  0 

27234.  2 

27243.4 

272'^2.  6 

27261,  8 

28.3 

28.4 

27271.0 

27280.  2 

27289. 4 

27298.  6 

27307.  8 

27317.0 

37326.  3 

27335. 3 

27344.  5 

27353. 7 

28.4 

28.5 

27362.  9 

27372.  0 

27381.2 

27390.  4 

27399.  5 

37408.  7 

37417.  3 

27427.  0 

27436. 1 

27445.  2 

28.5 

28.6 

27454.4 

27463.5 

27472.  6 

27481.  8 

27490.  9 

37500.  0 

27509. 1 

27518.  2 

27527.4 

27536.  5 

28.6 

28.7 

27545.  6 

27554.7 

27563.  8 

27572.  9 

27582.  0 

37691. 1 

27600.  2 

27609. 3 

27618.  3 

27627.  4 

1 

0.9 

28.7 

23.8. 

27636.  5 

27645.  5 

27654.  6 

27663. 7 

27672.  7 

37681.  8 

27690.  8 

27699.9 

27708.  9 

27717.9 

2 

1.8 

38.8 

28.9 

27727.0 

27736.  0 

27745. 1 

27754. 1 

27763. 1 

27772. 2 

37781.  3 

27790,  2 

27799.  2 

27808. 3 

3 

2.7 

28.9 

29.0 

27817.  2 

27826.  2 

27835.  2 

27844.  2 

27853. 2 

37863. 3 

37871.  2 

27880.  2 

37889. 1 

27898. 1 

4 

3.6 

29.0 

29.1 

27907.1 

27916. 1 

27925.  0 

27934.  0 

27943.  0 

27951.  9 

37960.  9 

37969.  8 

37978.  8 

27937.  7 

5 

4.5 

29.1 

29.2 

27996.  7 

28005.  6 

28014.  6 

28023. 5 

28032.  4 

28041.  4 

28050.  3 

38059.  3 

38068.  2 

38077. 1 

6 

5.4 

29.2 

29.3  ! 

28086.  0 

28094.  9 

28103.8 

28112.  8 

28121. 7 

28130.  6 

28139.  5 

28148.  4 

28157.  3 

28166.2 

7 

6.3 

29.3 

29.4  , 

28176. 1 

28184.  0 

28192.  9 

28201.  7 

28210.  6 

28219.  5 

38328.  4 

38237.  3 

28246. 1 

38364.  9 

8 

7.2 

29.4 

29.5 

28263.  8 

28272. 6 

28281. 5 

28290.  3 

28299.  2 

28308.  0 

28316.  9 

28325.  7 

28334.  5 

28343.  4 

9 

8.1 

39.5 

29.6 

28352.  2 

28361.  0 

28369.8 

28378.  7 

28387.  5 

28396.  3 

28405. 1 

28413.  9 

28J22.7 

28431.  5 

29.6 

29.7 

28440.  3 

28449. 1 

28457.  9 

28466,  7 

38475.4 

2848J.  2 

28493. 0 

28501.  8 

28610.  6 

28519.  3 

29.7 

29.8 

28528. 1 

23536.  9 

28545.  6 

28554. 4 

28563.  2 

28571.  9 

28580.  7 

28589. 4 

28598.  2 

28606. 9 

29.8 

29.9 

28615.  7 

28624.  4 

28633.2 

28641.  9 

28650,  6 

28659.  3 

28668. 1 

28676.  8 

28686.  5 

28694.  3 

1 

8.6 

29.9 

30.0 

28702.9 

28711.  6 

28720.  3 

28729.  0 

28737.  7 

28746. 4 

28755. 1 

28763.  8 

28772.  5 

28781.  1 

3 

1.7 

30.0 

30.1 

28789.  8 

28798.  5 

28807.  2 

28815.  9 

2S824.5 

28833. 2 

28841.  9 

28850.  5 

28859.  2 

28867.  9 

3 

3.6 

30.1 

30.2 

28876.  5 

28885.  2 

2B893.  8 

28902.  5 

28911. 1 

38919. 8 

28928. 4 

38937.  0 

28945. 7 

28964.  3 

4 

3.4 

30.2 

30.3 

28962.  9 

28971.  5 

28980. 1 

28988.  8 

28997. 4 

39006.  0 

29014.  0 

29023.  2 

39031.7 

29040.  3 

4.3 

30.3 

30.4 

29048.  9 

29057.  5 

29066. 1 

29074.  7 

29083.  3 

29091.  8 

29100.  4 

29109.  0 

39117. 6 

29126.  2 

6 

5.2 

30.4 

30.5 

29134.  7 

29143.  3 

29151.  9 

29160.4 

29169. 0 

29177.  6 

29186. 1 

29194.  7 

29203.  2 

29211.8 

7 

6.0 

30,5 

30.6 

29220.  3 

29228.  9 

29237.  4 

29245.9 

29254.  4 

29262.  9 

29271.  5 

29280.  0 

39283.  5 

39297.  0 

8 

6.9 

30.6 

30.7 

29305.  5 

29314. 0 

29322.  5 

29331. 1 

39339.  6 

29348. 1 

29356.  6 

29365. 1 

29373.  6 

29382.  0 

9 

7.7 

30.7 

30.8 

29390.  5 

29399.  0 

29407.  5 

29416.  0 

29424. 4 

29432. 9 

29441. 4 

29449.  8 

29458.  3 

29466.  8 

30.8 

30.9 

29475. 2 

29483.  7 

29492. 1 

29600.  6 

29509. 0 

29517. 5 

29525.  9 

29534.  3 

29542.  8 

29551.  2 

30.9 

134 


A  MAXITAL  or  TOPOGRAPHIC  METHODS. 


Taisle  II. — Correct  ion  for  r — r',  or  diffo-etice  in  the  temperature  of  the  barometers  at  the  two  stations. 

This  correction  is  neffative -when  the  attached  thermometer  at  the  upper  station  is  lowest;  po^tive  when  tlie  attached 
thermometer  at  the  upper  station  is  hip;hest.] 


Cor- 

Cor- 

Cor- 

Cor- 

Cor- 

Cor- 

Cor- 

Cor- 

Cor- 

F. 

tion. 

r. 

tion. 

F. 

tion. 

E.ft. 

F. 

tiOD, 
E.ft. 

F. 

tion. 

F. 

tion. 

F. 

tion. 

F, 

tion. 

F. 

tion. 
E.  ft. 

F. 

rec- 
tion, 

o 

E.ft. 

E.ft. 

E.ft. 

E./t. 

E.ft. 

E.ft. 

o 

E.ft. 

l.C 

2.3 

11.0 

25.8 

21.  C 

49.2 

31.0 

72.6 

41.(1 

96.0 

51.(1 

119.5 

61.0 

142.9 

71.0 

166.3 

81. (. 

189.7 

91,  t 

213.2 

1.5 

X5 

11.5 

26.9 

21.5 

50.4 

31.5 

73,8 

41,5 

97.2 

51.5 

120. 6 

(il.5 

144.1 

71.5 

167.  5 

81.5 

190.9 

91.5 

214,  3 

2.0 

4.7 

12.0 

28.1 

22.  C 

51.5 

32.0 

75.0 

42.0 

98.4 

52.  (1 

121.8 

(i2.  (1 

145.2 

72.  (1 

168,7 

82.0 

192.1 

92.0 

215.5 

a.i> 

5.9 

12.5 

29.3 

22.5 

52.7 

32,5 

76.1 

42.5 

99.6 

52.  5 

123.0 

62.5 

146.4 

72.  5 

169.8 

82,5 

193.3 

92.5 

216,7 

3.U 

7.0 

13.0 

30.5 

23.0 

53.9 

33.0 

V7.  3 

43.0 

100.7 

53.0 

124.2 

63.0 

147.6 

73.0 

171.0 

83.0 

194.4 

93.0 

217.9 

3.5 

8.2 

13.5 

.31.6 

23.5 

55.1 

33.5 

78.5 

43.5 

101,9 

53.  5 

125.3 

63.5 

148.8 

73,5 

173.2 

83,  5 

195.6 

93.5 

219.0 

4.0 

9.4 

14.0 

32.8 

24.  C 

56.2 

34,  11 

79.6 

44,(1 

103,1 

54, 11 

126.  5 

64,(1 

149,9 

74,(1 

173.4 

84,(1 

196.8 

94.0 

220.2 

4.5 

10.5 

14.5 

34.0 

24.  5 

57.4 

34,5 

K0.8 

+4,5 

104.2 

54.5 

127.7 

64.5 

151.1 

74,5 

174,5 

84,5 

197.9 

94.5 

231.4 

5.0 

11.7 

lo.O 

35.1 

25.  (i 

58.6 

35, 11 

82.  0 

45.0 

105.4 

55,  (1 

128.8 

65.0 

152.3 

75,0 

175.7 

85,0 

199.1 

95,0 

222.5 

5.5 

12.9 

lo.5 

36.3 

25.5 

59.7 

35.5 

83.2 

45.^ 

,  106. 6 

55,5 

130,0 

65.5 

153.4 

75.5 

176.9 

85,5 

200.3 

95,5 

223.7 

6.0 

14.1 

16.0 

37.5 

26.0 

60.9 

36,  0 

84,3 

46,0 

107,8 

5fi,  0 

131.2 

66,  0 

1.54.6 

76,0 

|'J78.  0 

86.0 

201.5 

96.0 

224.  9 

6.5 

15.2 

16.  b 

38.7 

26.5 

62.1 

36.5 

85,  5 

46,5 

108,9 

56.  5 

132,4 

06,5 

155.8 

76.5 

•179.  2 

r6,5 

202.6 

96,5 

226.1 

7.0 

16.4 

IV.  0 

39.8 

27,  t 

63.2 

37.  C 

86.7 

47,0 

110.1 

57.0 

133.5 

6f,  0 

157.0 

77.0 

180.4 

87,0 

203.8 

97.0 

227.2 

7.5 

17.6 

17.5 

41.0 

2V.5 

64.4 

37,  5 

87.  H 

47.5 

111,3 

57.5 

134.  7 

67,  5 

158,  1 

77.5 

181.6 

87,5 

205.0 

97.5 

228.4 

8.0 

IS.  7 

18.0 

42.2 

28.0 

65,6 

38,0 

89.  0 

48,0 

112,4 

58.0 

135.9 

68,0 

159,3 

78,0 

182,  7 

88.0 

206.1 

98.0 

229.6 

8.5 

19.9 

18.5 

43.3 

28.5 

66,8 

38,5 

90.2 

48,5 

113.6 

58.  5 

137,0 

68.5 

160.  5 

78.5 

183,9 

88.5 

207.3 

98.5 

230.  7 

9.0 

21.1 

19.0 

44.5 

29.  (1 

67.9 

39.0 

91.4 

49.0 

114,8 

59,0 

138.2 

69,0 

161.6 

79.0 

185,1 

89.0 

208.5 

99.0 

231.9 

9.5 

22.3 

19.5 

45.7 

29.  5 

69,1 

39.5 

92,5 

49.5 

116,0 

59,  5 

139.4 

(i9,  5 

162.8 

79.5 

186.2 

89.5 

209.7 

99.5 

233.1 

iO.O 

23.4 

20.0 

46.9 

30.(1 

70.3 

40.0 

93.7 

50.  0 

117,1 

60.0 

140.6 

70.  (1 

164.0 

80.0 

187.4 

90.  (1 

210.8 

100.0 

234.3 

10.5 

24.6 

20.5 

48.0 

30.5 

71.4 

40.5 

94.9 

50.5 

118.3 

60.5 

141.7 

70.6 

165.2 

80.5 

188.6 

90.  b 

212.0 

100,5 

235.4 

Table  III. — Correction  for  the  difference  of  (fravity  fn  various  latitudes. 
[  Correction  ^os^t(/«e  from  latitude  0°  to  45°;  negative  from  45°  to  90°.] 


Ap. 

Latitude. 

Ap- 

proxi- 
mate 

proxi- 
mate 

1 

diCfer-      JO 
ence  of  ana 
level,  l"" 

20'    40 

6° 

8° 

10°  12°  14° 

16°  18° 

20° 

22° 

24° 

26° 

28° 

30° 

82° 

34° 

36° 

38° 

40° 

42° 

44° 

45° 

differ- 
ence of 
level. 

88°  86° 

84° 

82° 

80° 

78°:  76° 

1 

74° 

72° 

70° 

68° 

66° 

64° 

62° 

60° 

58° 

56° 

64° 

52° 

50° 

48° 

46° 

Eng./t.  Ft. 

Ft.'  Ft. 

^ 

Ft. 

Ft. 

Ft. '  Ft.. 

Ft. 

Ft. 

Ft. 

Ft. 

Fl. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Fc. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Eng.ft. 

1,000  1  2.6 

2.6  2,6 

2.5 

2.5 

2.4;  2,4   2.3 

2.2 

2,1 

2,0 

1.9 

1.7 

1.6 

1.5 

1.3 

1.1 

1,0 

0.8 

0,6 

0.5 

0.3 

0.1 

0 

1,000 

2.000     5.2 

5.2  5.1 

5.0 

4.  9   4.  7    4.  6 

4.4 

4.2 

4,0 

3.7 

3.5 

3.2 

2,9 

2,6 

2,3 

1.9 

1.6 

1,3 

0.9 

0.6 

0.2 

0 

2,000 

3,  000  I  7.  8 

7.8   7.7 

7^6 

7,5 

7.  3I  7. 1    6.  9 

0.6 

6.3 

6.0 

5.6 

5.2 

4.8 

4.4 

3,9 

3.4 

2.9 

2,4 

1,9 

1.4 

0,8 

0.3 

0 

3,000 

4,000   10,4 

10,410.3 

10,2 

10,0 

9.8'  9,5    9,2 

8.8 

8.4 

8.0 

7.5 

7.0 

6.4 

5.8 

5.2 

4.6 

3.9 

3.2 

2,5 

1.8 

1.1 

0.4 

0 

4,000 

5,000   13,0 

13. 0  12. 9 

12.7 

12.5'12,211,911.5 

1         1 

11.0 

10.  5  10.  0 

9.4 

8.7 

8.0 

7.3 

6.5 

5.  7:  4.  9 

4.0 

3.1 

2.3 

1.4 

0.5 

0 

5,000 

6,  000    15.  6 

15,  6  15,  4 

15,3 

15.  0  14,  7  14.  3  13,  8 

13.2 

12.  6  11.  9 

11.2 

10,4 

9.6 

8.7 

7.8 

6.8   5,8 

4.8 

3,8 

2,7 

1.6 

0.5 

0 

6,000 

7,000    18.2 

18.  2  18,  0 

17. .« 

17..".  17.  I  16.616.1 

15,  4  14,  7  13,  9 

13.1 

12.2 

11.  2'10.  2 

9.1 

8.0   6.8 

5.6 

4.4 

3.2 

1.9 

0.6 

0 

7,000 

8,000    20.8 

20,  7  20.  0 

li'i.  :i 

Jii.  n  ]■."..-.]!).  0  18  4 

17.610.815.9 

15.0 

13,9 

12,811,6 

10.4 

9.1    7,8 

6.4 

5.0 

3,6 

2.2 

0.7 

0 

8,000 

9,000   23.4 

23.  3  23.  2 

_'J.  11 

JL'..VJ-J,  1121.4  20.  7 

19.  .S  18.  9  17.  9 

16.8 

15,7 

14.  4I13. 1 

11.7 

10.3   8.8 

7.2 

5.7 

4.1 

2.4 

0.8 

0 

9,000 

10,  000 

26.0  25,9  25,  7  ■_'.'..  4 

j 

2.-..  (J  24.  4  2:i.  8  23.  U 

22.  0  21.  0  19.  9 

18.7 

17.4 

16.  0[14. 5 

13.0 

11.4'  9.7 

8.0 

6.3 

4,6 

2.7 

0.9 

0 

10,000 

11,000 

28.6'28.5  28.3i28.0 

27,  526.  9  26, 125,  3 

24.3,23.121.9 

20.6 

19,1 

17.  6  16,  0 

14.3 

12. 5  10. 7 

8.8 

6.9 

5.0 

3,0 

1,0 

0 

11,  000 

12,  000 

31.2  31,130,9:30.5 

30.0  29.3  28.5  27.5 

26.5I25.223.9 

22.4 

20.9 

19.  2a7.  4 

15.6113.711.7 

9.6 

7.  5 

5.4 

3.3 

1.1 

0 

12,  000 

13,  000 

33.  8  33.  7  33.  5,33'.  1  32.  5  31,  8,30.  9  29,  8 

28.7:27.3 

25.9 

24.  3  22.  6 

20.  8:i8. 9 

16.  9il4.  8  12.  7 

10.4 

8,2 

5.9 

3.5 

1.2 

0 

13,  000 

14,  000 

36. 4  36,  3  36.  0  35,  6  35,  0  34.  2  33.  3|32. 1 

30.9:29.4 

27.9 

26.  2  24.  4 

22.4 

20.4 

18,2 

16.  0  13. 6 

11.2 

8.8 

6.3 

a8 

1.3 

0 

14,  000 

15,  000 

39.  0  38.  9  38.  6,38. 1  37.  5  36.  6  35.  6  34. 4 

33, 1I3L  6 

1 

29.  9 

28. 1  26. 1 

24.0 

21.8 

19,5 

17. 1J14.6 

12.1 

9.4 

6.8 

4.1 

1.4 

0 

15,000 

16.  000 

41.  641.  5 41.  2 4(1.  7  40,  0  39. 1  38.  0  36.  7 

35,  3  33.  7 

31.9 

29.9  27.8 

25.6 

23.3 

20,8 

18.  2  15.  6 

12  9 

10.1 

7.2 

4.3 

1.5 

0 

16,  000 

17,  000  144. 2 44, 1  43.  8  43,  2  42,  541,  5 40. 4|39.  0 

37.  5:35.  8 

33.9 

31.  8  29.  6 

27.2 

21.7 

22.1 

19.  4  16.  6 

13.7 

10.7 

7.7 

4.6 

1.5 

0 

17,  000 

18,  000    46,  8  46,  7  46,  3  45,  8  45,  0  +4,  0  42,  8,41,  3 

39.  7137.  9 

35.8 

33.  7  31.  3 

28,8 

26,2 

23.4 

20.  5  17.  5 

14.5 

11.3 

8.1 

4.9 

1.6 

0 

18,000 

19,  000    49,  4  49.  3  48.  9  48,  3  47,  5  40.  445. 1 

43.6 

41.9 

40.0 

37.8 

35. 5  33. 1 

30.4 

27.6 

24.7 

21.  7  18,  5 

15,3 

12.0 

8.6 

5,2 

1.7 

0 

19,000 

20,000    52.0,51.9  51.5  50.4  50.0 

48.  9147,  5 

1 

45.9 

44,1 

42.1 

39.8 

37.4  34.8 

32.0 

29.1 

26.0 

22.  819.  5 

16,1 

12.6 

9.0 

5.4 

1.8 

0 

20,000 

21, 000  '54.  eW.  5  54.  l'53,  4'52,  5 

51.  3  49.  9 

48.2 

46.3 

44,2 

41.8 

39,  336,  5 

33,6 

30.5 

27.3 

23.  9  20,  5 

16.9 

13.2 

9.5 

5.7 

1.9 

0 

21,  000 

22,  000   57. 2  57. 1  56.  6  55,  9  53.  0 

53.7;52.3j50.5 

48.5'46,3 

43.8 

41. 1  38. 3 

35,2 

32,0 

28.6 

2^.121.4 

17.7 

13.8 

9.9 

6.0 

2.0 

0 

22.000 

23,  000  '59.  8  59  7  59,  2 ,58,  5  57.  5 

56,  2'54.  6  52,  8 

50,7'48.4 

45.8 

43.  0  40,  0 

36.8 

33,4 

29,9 

26.2  22.4 

18.5 

14.5 

10.4 

6.2 

2.1 

0 

23,  000 

24,000    62,4162,2  61,8  61.0:60.0 

58.  6  57. 0,55. 1 

52.  9j50.  5 

47.8 

44,9  41.8 

38.4 

34,9 

31,2 

27.  4  23.  4 

19.3 

15.1 

10.8 

0.5 

2.2 

0 

24,  000 

25,  000  ,65. 0  64.  8  64,  4,63.  6|62.  5 

61,1,59.4  57,4 

55, 1  52.  6 

49.8 

46.8  43.5 

40.0 

36.3 

32,5 

28,  5  24  3 

20.1 

15,7 

11.3 

0.8 

2.3 

0 

25,  000 

BAEOMBTEIC  TABLES. 


135 


Table  IV. — Correction,  for- 


Decrease  of  gravity 

Decrease  of  gravity 

Decrease  of  gravity 

Approxi- 

Approxi- 

mate 

mate 

mate 

difterence 

difference 

of  level. 

<» 

+500 

of  level. 

O 

+500 

of  level. 

0 

+500 

Eng.feet. 

Feet. 

Feet. 

Fng.  feet. 

Feet. 

Feet. 

Eng.feet. 

Feet. 

Feet. 

1,000 

2.5 

3.9 

10,  000 

29.8 

31.5 

19,  000 

64.8 

67.0 

2,  UOC 

5.2 

6.6 

11.  000 

33.3 

35.1 

20,  000 . 

69.2 

71.4 

3.000 

7.9 

9.3 

12,  000 

36.9 

38.7 

21, 000 

73.6 

75.9 

4,  Olio 

10.8 

12.2 

13,  000 

40.6 

43.5 

2i,  000 

78.2 

80.5 

5,000 

13.7 

15.2 

14,  000 

44.4 

46.3 

23,  000 

82.9 

85.2 

6,000 

ie.7 

18.3 

15.  000 

48.3 

50.3 

24,  000 

87.6 

90.0 

7,000 

19.9 

21.5 

16.  000 

52.3 

54.3 

25, 000 

92.5 

94.9 

8,000 

23.1 

24.7 

17.  000 

56.4 

58.4 

9,000 

26.4 

28.1 

18,  000 

60.5 

62.6 

Table  V. — Correction  for  the  height  of  the  lower  station. — Positive. 


Approxi- 
mate 

Height  of  the  barometer 

in  En 

jlish  inches. 

Height  of  the  barometer,  in  En 

glish  inches, 

at  lower  station. 

mate 

at  lower  station. 

of  level. 

16 

18 

20 

32 

34 

36 

38 

of  level. 

16 

18 

30 

32 

34 

26 

28 

Eng.feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Eng.feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

1,U00 

1.6 

1.3 

1.0 

0.8 

0.6 

0.4 

0.3 

14,  000 

21.9 

17.8 

14.1 

10.8 

7.7 

4.9 

3.3 

2,000 

3.1 

2.5 

2.0 

1.5 

1.1 

0.7 

0.3 

15,  000 

23.5 

19.1 

15.1 

11.5 

8.3 

5.3 

3.5 

3,000 

4.7 

3.8 

3.0 

2.3 

1.7 

1.1 

0.5 

16,  000 

25.1 

20.3 

16.1 

12.3 

8.8 

5.6 

2.7 

4,000 

6.3 

5.1 

4.0 

3.1 

2.2 

1.4 

0.7 

17,  000 

26.6 

2L6 

17.1 

13.1 

9.4 

6.0 

3.8 

5.000 

7.8 

6.4 

5.0 

3.8 

2.8 

1.8 

0.8 

18,  000 

38.2 

33.9 

18,1 

13.8 

9.9 

6.3 

3.0 

6,000 

9.1 

7.6 

6.0 

4.6 

3.3 

2.1 

1.0 

19,  000 

39.8 

34.1 

19.2 

14.6 

10.5 

6.7 

3.2 

7,000 

U.O 

8.9 

7.1 

5.4 

3.9 

2.5 

1.2 

20,  000 

31.3 

35.4 

20.2 

15.4 

11.0 

7.0 

3.3 

8,  001) 

13.5 

10.2 

8.1 

6.2 

4.4 

2.8 

1.3 

31, 000 

33.9 

26.7 

21.2 

16.1 

1L6 

7.4 

3.5 

9,000 

14.1 

11.4 

9.1 

6.9 

5.0 

3.3 

1.5 

22,  000 

34.5 

28.0 

22.2 

16.9 

12.1 

7.7 

3.7 

10,  000 

15.7 

12.7 

10.1 

7.7 

5.5 

3.5 

1.7 

33,  000 

36.0 

29.2 

23.2 

17.7 

12.7 

8.1 

3.8 

11,000 

17.2 

14.0 

11.1 

8.5 

6.1 

3.9 

1.8 

34,  OOU 

37.6 

30.5 

34.2 

18.5 

13.2 

8.4 

4.0 

12,  000 

18.8 

15.3 

12.1 

9.3 

6.6 

4.2 

2.0 

25,  000 

39.1 

31.8 

25.2 

19.2 

13.8 

8.8 

4.1 

13, 000 

20.4 

16.5 

13.1 

10.0 

7.2 

4.6 

2.3 

i 

136 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


t5 

n 

^'' 

s 

i  ^ 

1 

o 

1- 

^ 

U  - 

H 

s 

>" 

^ 

'3/s 

.a 

■9. 

^ 

S.S 

^ 

11 

» 

S 

iii 

•7 

1 

Sg'-E 

1— I 

5  p..!: 

H5S 
3  a 


i 

s 

0 

;  ;.n=.a. 

C»00^C..O 

ooo^^co 

M<n— too 

mSSSc5 

:?ggBS 

.    .eD-*03 

ciNCJCJr^ 

rtrtrtrt 

CS 

:  ic^cQ- 

co-«cqca-i 

:"3:^" 

c<.-,o§3 

I-i  rH  i-i 

OOfMOiO 
OOOQOC-C- 

fSgSgg 

s 

:  irtoo 

oocoooo 

l>lO-#CO(M 

rH  r-tOOO 

£5S£^?2 

ggggg 

:   ;="*" 

MWMC^T-i 

rHr-.^ 

t* 

I      !os-*U3 

ai»r5(MCsoo 

oiflcococq 

^oSSSg 

2*00030 

SSoSS 

50 

;    ioca^ 

QO-^rHOJ  t- 

in-*D3MrH 

rtoSSS 

oot-c- t-co 

SoSSS 

.      .in-^ro        Ncq<Mr^rH        r-lrtrlrtrH        rt  rH                                                                                                   | 

§ 

;      i^rfCO 

t-cooooco 

in^WNrH 

oosmooS 

Sflpgsg 

ggggs 

:  :"=-*« 

WWci-^r^ 

■-1 

-* 

:  :«c^ 

0(MOt-<S> 

'^COM^O 

§g£Sg 

gE=SSg 

ssssg 

■    .loi^eo 

g 

■      'Ot-O 

l.O,-i«t-lD 

^(NW^O 

C5  00C0C-C~ 

t-o  000 

SSSgg 

Cicq^r^^ 

S 

-^OOOO-* 

«M^  =  § 

gsgeg 

SSSSK 

InSio^^ 

.     .  -fj!  PS  cj 

cici  —  r^rH 

^^^^    • 

s 

•  oo:Ot*  t- 

<M_^0§g 

ggggg 

SSSfelg 

5?S§^1§ 

g 

L-s  cocao 

WCO^CO 

<M^og£ 

w  t^wciin 

gSKSS 

000  t-lfiiCO 

10  -#"*■* -^n 

s 

:CQ.H-<U5 

-iOOO^M 

-ioSooco 

5!S3S5l 

■SO-^COM 

<-4  ^  r-I  eH  .-4 

--I  ^    *    '    ' 

ce 

■  cnocsco 

OJi>iacoM 

000"<i<00 

cooiincaos 
t-oocoin 

sssss 

SSSSra 

t* 

i--»- 

»°--". 

0  CI  00  I-  l> 

gg^sts 

sss^^ 

CO— (OQO  t- 

.*Ti..a.coco 

9 

;-.--.- 

-;-;-;-;-; 

gSoS^g 

0  JoStnS 

sss^^ 

005  treses 

S 

icscQ^c; 

o^co-^g 

ca  .-H  ITS  0  10 
0000  t-t-eo 

cOinirairs-* 

J;3iS5!g 

(KOH.Tl.M 

■  '#(rONr-5 

^rHi-li-H 

" 

leooMoo 

«,CO^OS 

sssggs 

l>-*.-H30<D 

eowocoo 

"*■*■*  CO  CO 

mMcoeoM 

« 

■NOOi-t  t- 

„CQ-.SS 

tr-  ,-1  uO  0  ttJ 
C-|>toOiO 

sg5;3S 

OOOt-lO-# 

"t  CO  en- CQ  so 

sasss 

.■*!M(M'-( 

tAAtA 

s 

COr-.C!aot- 

c-  0  0  0  in 

SSSSg 

55Jg^g=^ 

gssss 

■  CQM'-ItH 

r-.  r-i            *     ' 

- 

(MtO^CO-* 

MoS»P! 

ggsss 

sgs^g 

'JiCOiHOO 

sasssi 

t-COlM'-i.-i 

0 

lflCO(M«;cO 

^gsgg 

gssss 

'^  CI  CO  CO  « 

^  000  t-to 

s^ssa 

* 

ornoioN 

g'asss 

sssss 

conSSS 

I5S88S 

ssssgs 

00 

QO  l~  0  10  10 

5;5sss 

sssss 

^asss 

sssss 

t» 

0rt.n-.5J 

gSL^^S^ 

— >QOtn(ro  0 
"*  CO  com  TO 

SSSjSSs 

EJS2SS. 

SS3SS 

« 

ocsco  c:  t> 

So^"m 

gggss 

§i?3i3SS 

000  t-oo 

S33SS 

0 

COfflrtSS 

•rftL Its  CO 

sssss 

SSSSS 

C5-*..*  ^CO 

sssss 

c^c<;rJ  •  •      ■  ■  : 1 

■* 

ccoSgS 

"<JlMCO(M(M 

MCJMr-i-H 

SSSSS 

sssss 

ooggg 

« 

ojSS^g 

ssssss 

22233 

lM.-irt  0  0 

gggSS 

sssss 

« 

rtiSSSS 

M  Ci  (O  -*  CO 

SSSgg 

00000 

ggggg 

00000 

- 

§g?§3SS 

.-ic:oo  t^c- 

r-lC  0  00 

00000 

^s^gg- 

00000 

sgggg 

1 

<-o1 

Ssoo© 

:sr,xc:o 

^«M*« 

e  t-  X  ft  « 

SSPSS 

ei-CC»3 

ALTITUDE  TABLES. 


187 


4i 
O 

1 

R 

o 

■ntoromci 

■*03(MrH0i 

cnooc-oco 

in  ^  Tti  CO  CO 

COCOCONCN 

wcawNr^ 

®       ; 

^U=  00  CO  00 

■^  roiH  o  ci 

ooc-c-a=in 

in^^cQCO 

»in-*-<»m 

COCO(MM(M 

(Mrrqcq(Mi-i 

«       i 

corf  t^cqoO 

■^iMi-lOCi 

oot-omin 

^^COCQCO 

CO  in-*-*  CO 

cococqNN 

c4wNM^ 

t*           ; 

MCO  »'-(  t- 

2.3 
2.2 
2.1 
1.9 
1.9 

00  t-  CD  in  in 

^^CO«N 

*           1 

;   ; 

WNCOOCO 

COOOOCr. 

Mi-tOOlOO 

L-  t-  o  in  in 

^cooiroN 

,om^-*oi 

comcJ^ei 

(M*  N  W  r-i  rH 

ta 

Orinoco 

CDUJ-JHirf 

(M  r-(OC35CO 

^22";^ 

3 

oorBoin 

Noc-inco 

C^.^ffiO0  00 

t-coininrl. 

«coco«cq 

iriirirjcoco 

COC<lSQr>« 

S 

"T" 

t-acooon. 

MOCJQOt- 

CO  CO  in-*..* 

MCONM-H 

■"■*-*  CO  CO 

oJoacqoiN 

(MMi-lrH  t-1 

- 

S2SSS 

riSS^S 

rHOOOOC- 
Cd  W^^r-^r-^ 

"IS^" 

SS2S!^ 

s 

lot- '-it- CO 

ITS-*-*  CO  CO 

ooom-#iM 

rHOoo  c-t- 

""!^23 

^^^;i;i 

« 

iri  rh  TlH  CO  CO 

o  t-incoiM 

oosooc-eo 

!!^"2!^S 

(MfMNiHr-l 

o 

cocooicsca 

ww(^it^iN 

c^  OS  cot- to 

in  r»  rK  CO  CO 

(MCa«H.Hr4 

r-i  r-(  iH  iH  fH 

-*- 

"T~ 

:  : 

COO^MrH 

OiOOt-COCC 

in-*r}ico(n 

(M  (M  iH  r-l  O 

in  r»  CO  CO  CO 

CJNNMM 

rfrlrHrfrt 

^ 

"* 

r-l-*  00-*r-( 

Ci  (?i  (N  (N  ci 

S!^"3!^ 

!4Z;!^^3 

sl 

OCOC^POO 

t-incoi-io 

ffiCOt-OiA 

rJlritCOIMM 

r-ti-CrHOO 

inriicococo 

Mwcacqw 

r-ljH  T-l  rH  rH 

"*  i 

en  (Mc- CO  OS 

t--^(nrHCS 

00C-(D  lOO 

r,com«(N 

iHr^OoS 

MNWW-I 

rHrHrH  -H 

^! 

oorftoiDcm 

oot>  to  in-* 

rHiHOOJOS 

ni^-cioici 

(NcdMfN  tH 

" 

SciMWr^ 

----- 

cococgc.^. 

rlO  OClOi 

■* 

tsojrjoc- 

t-to  m"*-# 

CO(M(M  rHr4 

O  O  CT.  O)  0> 

^cicicoN 

cq  (M  (M  i-l  .-I 

3 

raioocooi:- 

oco  in-*  CO 

COIN  rHrHiH 

ogSSS 

© 

M  t-MCitO 

■^caoojc- 

.in^^co 

Mcqr-tiHO 

OOlOOiQO 
rA      •      ■      '      ' 

i- 

— 

<d_>ri_ 

COrHCnOOC- 

tH  rH  fH  rS  rH 

--.--  = 

^^^^^ 

r-(lOrHt-»Cl 

Mr-.  0=00(0 

oin^coco 

<MrHi-lOOJ 

gs§§ss 

t=^ 

Tj5  CO  cowed 

C4  eg  J  rH  rH 

■H  rHiH  1-" 

S 

oco 

OTHOt-rH 
-jcicriNCd 

C-icqr^r^r^ 

1-1  WO  OOT 
iH  rH  rH  r-i 

ffiTOOTOoS 

u 

M  rH  -;  rH  r^ 

"2"2!1 

iHrHrH 

C=>  t-'*rHQO 

>o 

cooioicJca 

rHCSt-K)U3 
CQrHrHrHrH 

tHOOOIO) 

g§sssg 

■* 

b: 

t-wcoinC3 

c4r-5  i-H  iH-i 

^2"S^ 

O  OOS  OSOO 

ooooc-c-c- 

» 

I-*co 

CO.H  t-rilrH 

3S""2 

rtrtrtrHr.; 

o§gg§ 

coot--it<M 
oooo  t-c-t- 

7 

loJM 

inotocorf 

o  t-cD  in-* 

CO  CC  0,^0 

ggssss 

O  t-^Mg 

.«5-* 

oic^NWN 

1-1 

CO  S  cd  eg  cq 

00  t-ic-^m 

COMiHrHO 

i-H  rH  ,iH  r-5  r-i 

g^ssss 

oomcM  or- 
t-c-t-c-co 

- 

« 

3o« 

I3§ 

c9i>xeaO 
OOOOrt 

iHeiM-*»o 

COt^oCCSO 

sijsas 

is;^iS 

138 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


i  1 

SSSKS 

ssss^ 

35!S33 

!M(Mr-<00>        0000C-(OlO        ift  ■*  ;2K  M  «         1 

-^  _,,  ^  ^Tj"  CO         CO  CO  CO  CO  CO         COCOCOCOCO 

1 

O) 

SSoS^ 

ggS55 

".^.^.^^. 

5!5SSg    S^SSS    SSSSS 

s 

gSlgiSS 

r-tOSOOOOCO 

^"^^^ 

3sS5§    SSSSS?    SggSS 

5    1 

s  s  s  ^  s 

SS5'53: 

SS5i3g 

gSSSS    SSS5SS    gSSSS 

S 

5;§!S3?I 

qJ5g^S 

SSSSS3    fSS5S?SS    SgSSSS    1 

iS    1 

CC  ■^  CS  CO  t- 

S33?J5; 

sggss 

gSSSiSSl    SSSSS    gSSSa    1 

» 

s§sss 

gg^ss 

SSSm" 

SgSS?S    SSSSS    SS5S§ 

M 

55!§33 

gssss 

sss^s 

gissss  §ss5?sa  assss 

1 

> 

•1 
i 

s 

u  1 

§§3^51 

o  =:  oo  t^  --D 

SISSSS 

sssss  sssss  sssaa 

»H 

5335S 

ssssa 

SSS3=3S 

ggSSS    SSSS8    33^SS    1 

o 

ssssss 

sssss 

SSSSS    SBSSS    SSSSS 

3S?SSS 

OOJCIXC- 

SSSSS    S33S=3    SSSSS 

IH 

COI>S>-#CC 

M  N-JO  OS 

sssss 

'tSSsass    S3Sg38S    SSgSS 

t* 

^^^^^ 

sssss 

sssss 

SSSS3S  ?3SE3a§  ssssa 

sggss 

§S6;S§ 

ss^ss 

SSaSS    SSS3S    sssss 

S3SgSS 

SSSS3 

assss 

^t-JOOO  ososxxoo  t-t-ot-co  1 
(MlMWrJW        ^,H.-t.-lrt        rtr-ti-liH-H 

3 

sssss 

SS3S3S 

ssaas 

SSSSS    33SSS    SSSS3 

S     1 

sssss 

assss 

SSSS3 

oooocot-t-      t-«coeo®      St233^ 

3   ! 

S3a_?5?3 

g3S?i§§ 

OOJCOOOt- 

S^3S3    S3S33    33333 

1-1 

SSI3S3S 

gasss 

t-t- t- jOffl 

SSSI33    33333    SSS33 

e 

SSSSS 

ooino^ 

33353    33333    3^333^ 

e> 

SSSSS 

sssss 

33333 

CONMMfM        iH  rH  i-i  i-H  1-4        OOOOO 

"  1 

t-(DCDif5in 

32333 

3S33S    S°S2g    goooo 

" 

O  -S"*  CO  ro 

SSS3;^ 

3;:1S33 

33g§8    ggooS    °So=° 

« 

sa2;^s 

S=l33a 

S8§§S 

ggggg  ggsss   sssss 

^OOOffl 

ggssg 

§s§ss 

SSSSS    ggggg    ggggg 

-* 

sssss 

ooooo 

ssggg 

ggggg   ggggg   ggsgs 

« 

0000  = 

ggggg 

in  mm  ■* -* 

ooooo 

3SSS3    SSSSS    Sgggg 

01 

ooooo 

gssss 

S§§go 

ggggg  ggggg  ggggg 

ooooo 

ggggg 

ggggg 

OOOOO      ooooo      ooooo 

■pi 

~ 

^^UZ'M 

i5i§i3 

^  ei  es  <«  u3 

5St-XCS©       i-cMM-*'©       '*'^SS* 

ALTITUDE   TABLES. 


139 


» 

03CqCQ-^rt 

sgss» 

00  00X00  t- 

i>.m'*(Mi-( 

^gsgs 

^H^,-(^ 

» 

!N(N(Nrtrt 

fHOOOJOS 

S3SBS 

-*MO00t- 

gsggs 

M^t-I 

s 

««rtrt« 

oogSS 

gssss 

gggsg 

t-i>c-t-to 

sggss 

rtrt 

5 

CjNrfrtrf 

ooSSS 

mSwSSo 

00t-C-t-E> 

rt  .-t  o  <n  t- 
t- t-c- toco 

ggsss 

rfrt 

O 

<N<Nrtr1  = 

gssss 

c^t>  t^  t?-^- 

sggss 

gggss 

r-,Wrfrtrt 

'-' 

:i 

=^-H«00 

oSSgg 

ssssss 

l^-l^-t- t-c- 

gggss 

ssssg 

--"-^-^         - 

■4H 
id 

sssss 

toincaooo 
00  0)00  00  c- 

SDiQcOtMO 

t^  t-t-t-c- 

ggggg 

ssggg 

« 

i-H^OOOi 

OSOlSoOOO 

sssss 

iococaocj 
t-  t- 1-  t-  to 

CO  to  coco  to 

sssgs 

s 

i-li-H  OOOl 

mffiSocS 

OoSc~  t^t- 

-*(M  rHOl  OO 

c-i-c-<o  to 

to  in  -*)  CO  (N 
to  toco»  to 

sssss 

s 

rtOoSS 

gssss 

OOQO  t>  t-C- 

CM  W  «  00  to 

t- t>co  to  to 

g^§5§ 

gg£g3 

o 

ooSSm 

gssss 

ssgse" 

t-(oco  toto 

^COi-COOJ 

to  CO  CO  com 

SKSg;S 

ooosSo) 

gssss 

t-c-c-c-r- 

ggggs 

ssggs 

SgSioS 

? 

oSSgS 

t-  -t  (MOOD 

CO  00  CO  00  t- 

tOTO  to  toco 

sgsss 

ggsss 

■J 

■* 

ggsss 

SS53?2g: 

mas  WOOD 
t-t-c-r-o 

t>m-*M.-i 

ggsss 

ssssa 

% 

■* 

ssggs 

Qo3£^t2 

c2t^?wS 

ssssg 

gssss 

sssss 

s 

-* 

gggss 

ooc- t-^-o 

f^ggSS 

sssss 

5SS3S 

SSSg5 

1 

-* 

ggsss 

o  t-  in  TO  (M 

00  t-  C-  t-  L- 

^^£oo 

S»SS5 

ssass 

3u?§5S 

fi 

-* 

ggssss 

C~I>  t-t-t- 

sssss 

sssss 

sssss 

sss?;!; 

-* 

00  CO  00  00  t- 

C-C^  t- tr- to 

SJS^gg 

sssss 

asssg 

OiOOO  to  to 

-* 

oooo«)^>^- 

^wggS 

sssss 

sssss 

cq.-(ooioi 

S55S3 

© 
-* 

ooaot-oD- 

t>t>eo  sous 

»SoSS 

SiSSSS 

ggSS5 

5!§S33 

» 

(MOi  t-irt  CO 
CO  t-c- t-^- 

t-(OOCOCO 

sssss 

SSSS3 

OOS  00  t-o 

^S33S 

s§ 

gssss 

sssss 

j^fOMOOS 

tH'*-*'*^ 

5:33^5! 

« 

t-C-t>C-CO 

sssss 

SBSSS 

S3g§3 

5SS3S3! 

COi?l(M.-IO 

i 

t>t>t>COX> 

ggssg 

^«5lg«M 

sssss 

IS^^SS 

Mr-HOOCl 

s 

'^.-laic-io 

sssss; 

SSSSiS 

55555 

^35^51 

5lSgSg 

•* 

sssss 

(Mooor-in 

S5SS3! 

CO  CO  (M  .-HO 

0  a)Q6QO^- 
-*cococort> 

u 

ggggg 

gS5SS 

sssss 

5SiS31S 

(N  WiHOO 

sssss 

f? 

ssggs 

sssss 

f-iO00t>«S 

!§:J35^ 

^^s^s 

ss=?ssg 

(H 

gssss 

SSSgg 

oi  00  t-om 
-Jl -^ -* '^  -11 

^-gg^s 

®g£gt;5^ 

sssssss 

■3^.2 

SSiSS 

mmmmS 

32S53S 

sss:ss 

140 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


ill 

^ 

.  o 

a 

OOJ 

- 

1-C 

o 

OS 

;  :  :^^ 

OS 

III!!         1     1  oojOi 

iS 

;  :  :  ;  :     ;  ■^^^ 

£ 

■  ;  :  ;  i     \°^^^ 

$ 

;  ;  ;  ;  :    oSSSS 

!S 

!    ;    1    !o 

§SS3§ 

;  ;  ;  1'^ 

-* 

:  ;  loOT 

gssss 

.      •       ■  1-4      ' 

SI 

!;  i^^ 

£SS§S 

Si>»eBO   iH(NOS-H»i 


ALTITUDE  TABLES. 


141 


a 
a 

a 
S 

s 

i 

cqesiwwo 

nSScScS 

oot-t-<oeo 

§SS5Sn 

^siass 

SSSnS 

CS 

i-IO  0003 

SSS5S 

§:;§SSS 

sssas 

sssssg 

c3!3Sc3c3 

1 

g 

OOSOiXIQO" 

sssss 

^SSSS 

aassss 

WMWWN 

sssss  1 

■  1 

s 

(NM(MM(M 

sgsss 

saiass 

SSS?3S3 

?3S;;ss 

sgsss  1 

1 

» 

sssss 

sssss 

S3SS5S 

I3?3S?,SS 

cSSSSS 

sssss 

g 

ssssss 

cj  w  wcg  w 

?5i58?3?3 

SSSS^ 

gssss 

ss3sa 

1 

-4* 

ssssa 

3c5"c5c3 

SI35SS 

SSS^S 

35333 

S3SSS    1 

es 

S33SS5 

S3S?3SS 

SnSSS 

O  CiCiOO 

00  OO  00  00  00 

t-c-t-c-c-      1 

S 

SSg^SS 

SSSnS 

SSS3S 

SS323 

00  t-t-t^t- 

sssss  1 

iH 

saass 

sssss 

sssss 

oocooot-t- 

c-rr-totoo 

OODWU5        1 

§ 

W^T-O    O 

OOOJOSOS 

OOOOOOOOC- 

c-c-t-o» 

oot=o«)' 

.nu50U5^        1 

S 

g°  =  SS 

OJOOOOCOtO 

c-c- t-t-o 

<o  o  o  o  .n 

^  o  ,n  U!  w 

„^^^^    1 

" 

sssss 

OT  t-c~c~r- 

^33SS 

3SS3S 

-*■*■*-*'* 

33333    1 

S 

OOCOX  t-l> 

EiSSSS 

OOWiOiO 

33333 

33333 

33333    1 

CD 

L-  t>  t>  m  «s 

ocgminift 

S3333 

33SS3 

33333 

33333    1 

2 

sssss 

SS333 

33333 

33333 

2333S 

s;J;iss 

« 

3S3333 

33333 

SSS2S 

3S333 

■-(  oooo 

323S3 

33333 

333=1  ;j 

OOOOO 

ooooo 

S3 

3S3S2 

33p;S=! 

SlUSSS 

33333 

ssg§§ 

ooooo 

s 

'-tt-^-'OO 

OOOOO 

CSCSCIOSO 

ooooo 

ooooo 

ooooo 

° 

woooo 

sasgg 

ooooo 

ggggg 

ooooo 

ooooo 

o 

S§g§g 

ggggg 

ggggs 

ggg  =  = 

ooooo 

ooooo 

00 

ggggg 

ggggg 

SS5S& 

SSSS5 

ggggg 

ggggg  1 

r>- 

sssss 

ooooo 

§oo§o 

ggggg 

ggggg 

ooooo 

» 

(o  aso  IT) » 
oo  oo  o 

ggggg 

o  o  lO  o  ira 
ooooo 

ggggg 

ooooo 

S3333    1 

« 

sssss 

irt  in  in  o  in 

ooooo 

irt  in -n -* -:}< 
ooooo 

ooooo 

ooooo 

OOOOO 

-^ 

33333 

33333 

ooooo 

ooooo 

corowffoco 
ooooo 

OOOOO 

=, 

gggSo 

ggggg 

ooooo 

ooooo 

ggggg 

ooooo 

« 

ggggg 

ggggg 

ggggg 

ooooo 

ooooo 

g3S33    1 

IH 

ooooo 

OOOOO 

ooooo 

ooooo 

ooooo 

ooooo 

Ms  .2 

^ 

osSss 

S2233 

Or^oOCSO 

i 


142 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


1' 

.g 
S 

o 

eoS o  SS 

^JStoSS 

sssss 

oSS^S 

s^sss 

!SS33IS 

s 

sgsss 

SSSfgSS 

3SSS3 

ing^^^ 

ft5;SSi3 

!g35S3 

s 

as  S«3iOlfi 

t-  » so  in  "* 

SgggS 

SS^^IS 

t-eoioiOTii 

^ssss 

1 

r- 

s§gss 

sssss 

gggg?! 

OSOOOOl>«S 

SSS33 

CTSCOW  W  i-H 

1 

?o 

eoioiretoin 

sssss 

sssss 

^ssss 

!§33!33 

^^^^51   1 

1 

§ 

irsinioioio 

OOOTMM 

S;5:S!S!S 

^31:515139 

^^5:^^  1 

1 

■* 

SSSISS 

sssss 

gg^ss 

:gS!g«^ 

55^35! 

■^-di-SI  MCO 

g 

BSSSS 

ininioic5-<* 

OSCDC-t-CO 

-*  -*  T(l  -tj*  T(» 

-*T(I-*^'* 

Sg^5i5! 

ssssg  1 

S 

sssss 

ifiioin-*"* 

S5;SS!S 

.^^^gg 

NWi-lOO 

gssss  1 

i-< 

SS3gS 

OOOiCC  t- 

5iSI5!§3 

^^gg^ 

55§§g 

gSSSR     1 

o 

gssss 

^^^^^ 

sss3;s 

^^"^^^ 

O  OOl  OS  00 

mmSmS      1 

2 

gSSS" 

^^^^^ 

i3:3!tS3 

gSmmcS 

sssss  1 

'^ 

gggss 

gis^gg 

^— '  OOOi 
-i*^-*  -t  CO 

SSSSS; 

ssssss   1 

t» 

o  oim  ooc- 

ss«g!5 

3^^551 

«om^oo 

SSKSS 

giSSoSS    1 

03  QO  O)  i^  eD 

-di 'S*  ■*  ■* -^ 

;3S333 

^s^^^ 

SSSmn 

ssssss   1 

5 

^^^^^ 

3353^ 

5:5S5g 

OCO  CO  t>l> 

SScttoS 

^^comco 

■* 
-* 

5;SSS3 

53^95! 

OOCIOIOO 
■* '3' CO  CO  CO 

sssss 

ssgss 

SISSS?3    1 

■* 

^!§:S3S 

OJ(N.HrHO 

ggggs 

£gS!§g 

gSSg?S5 

S5SSSE3     1 

5! 

53SgS 

.-HiHOOCi 

SSKSS 

sssss 

Sg?SSS3 

5?SSSg    1 

■* 

5!3ggq! 

^^^nn 

SSSgg 

^cSmmco 

SS?S(oS 

sssss   1 

■* 

3^5!  5!  51 

ss^ss 

ssssss 

SmSSS 

ggsss 

gggss  1 

» 

gssss 

oooo  t- t-eo 

eo  in  lo  ■*  ■* 

COCOCCTOCO 

iSg5SS?3 

53:;^5^gg 

g§§S?i 

^ 

5JSSSS 

t-  |>  CO  *o  »ft 

C-S  CO  TO  CO  03 

SS3S8 

sssss 

ssssp 

SSSncS 

S 

gssss; 

sssss 

mSiomm 

l^^c^^^ 

ggsss 

ssssa  1 

50 

JS^SSS 

sssss? 

coTOComco 

r-(00  OOl 

§SSSISS 

sssss   1 

sg 

t- tr-(0=DlO 

SSSSIS? 

ssssg 

sgggg 

ssssa 

sssss  |- 

-* 

sssss 

assss 

mSSSS 

sssss 

5SSS8 

M^f^cgw 

SSSSSo 

sssss 

S8SS" 

?sgsss 

a§s8s 

Cq<N  M(N  CM 

ssssgg 

raSSmS 

§Sg§?S2S 

assss 

sssss 

^^^^g^ 

s 

sssss 

^OOOSOl 

ssssss 

sssss 

ssssa 

§3S5?5?5g3    1 

° 

i<  -ti « -If  To 

ooooo 

S22;5S 

»t>.QDC50 

rH 

ALTITUDE   TABLES. 


143 


i 

S8g§§ 

00  00  OO  00  00 

oooooot-c-      c-t-t-tr-t- 

JiE^pgg    SSSSS 

» 

OSCiOsOCO 

§8SSSS 

(Momooi-      oin-#eo(M 
oooot-t-c^-       t-Ot-C-C- 

c^c^gSS    S£Sw» 

38 

sgggs 

ooSotooS 

oococ-t>t-      t-t-et-c- 

SgSSS    &Sg33 

1 

S 

SSSSS 

OCt>C-C-t>        C--t-t-t-l> 

ggg££    ggSSS 

1 

i 

goggg 

sssss 

esoot-i^o      lo-d'*!^*^ 

1 

a 
a 

1 

1 

s 

1 

s 

3SS8SS 

ssssss 

c-c-t-c-t-      c-t-t-t-o 

gSBgg    ggggg 

■* 

SSBSS 

sgsggg 

l>OU3-*CC        (NtHOmoO 

g£ggS    gSggg 

ss 

SoOWWOO 

sssggE; 

OW-<*iroM        T-(OOJOOOO 
t-l-l>I>t>       t-t-ooo 

£§§33    ggggg    1 

s 

ooSooooS 

i-(0  OJI>0 

00  OO  C-  t~  c- 

t— t-t-t-C-        L—  OOOO 

gSSoS    SSSSS 

s 

ggsss 

ociootD  If; 

00  t-t- t- t- 

aSSE^g    8g§£S 

oeoeotoeo      o  to  o  m  U5 

o 

S3§33§ 

cjoo  t-in  -* 
t-t-t-t- tr- 

TO^i-tog   SS&wS 

sssss  sssss 

Socooccc- 

ee  t-  to  in  D5 

t>t- 1- 1-  t- 

?2f;egg  g£sgs 

ggggg  ggsss 

£ 

SSS53^?2 

gE=gg§    SS§3S 

COMrHOO        OlOOQOt-O 

K 

SSg^^ 

t>  L^  t-  t-  t- 

.-(ocioot-      oin"*Mco 
t-t-ooo      <»oooeo 

jrarHOOO        '^^^S'nS 

1^- 

jHOmt-O 

t-  L~  b-  t-  t- 

ggg£g    S^SSSS 

Sg-ggg    SISggS 

g 

St^tS^ti 

SSSSig    SSSSs! 

oocsoot-      t- o  in  o -* 
otDioifjin      inirsioirsin      | 

t-t-t>r- t> 

SE2Eig§ 

oooSo      ooSoS 

SSSSS    SSlgiSiS 

S 

t-  c-oc-c- 

t^pI^SS 

5§gSg    SSSSg 

COOt-t-O         IftU^-JCOCO 

l> 

t-t- t-  ^>lr- 

Elgggg 

gg^So      oooSS 

OOt-OtOuS        -tJ-^COf^tN 

t- 

I>  t- 1*  t>  c- 

gssss 

sssss  sssss 

mmmirain      loioioinin 

o 

t-t-r- t-c- 

sssss 

T#C01MWi-l         OOSCOOOt- 

ocDootD      omininm 

gssss  sssss 

ffl 

t-c-t-t- to 

SBSSS 

C:)C<lM.HO        oioooot-o 

ifSift-*COCC         (MC-l'-HOO 

S 

c-t-  c-s^J 

BSSSS 

iMTHi-ipcj      QOi>i>oin 

igasss  ssggs 

(M  o  c:  00 1- 

1>  l-  (D  ^  to 

g£SSS 

358gg    KKSStS 

E 

OOlOOgg 

ssggs 

^ggss   E^ss;s:S 

COCOMfHiH         OOiClOOOO 

ifiininmifa      m -^  ■*  ■* -* 

» 

ggSSig 

sssss 

oSSSS      SSooEra 

CqWr-lOO         CiOlOOOOD- 

iniftirainm      ^jf  "<r  ■* -^  ■* 

•* 

SS^SJg^ 

sssgs 

OJQOD^Og        Ig^SSSs 

5(000® 

ggsss 

inmmifsm      moiniom 

SSSSS    55SSS 

gs^^ss 

rH  OCs  00  CO 

lr-0»fti.0'*       ^""SlftS 

g§555    5;^§iS!g 

g^ggg 

SSSSB 

ssssg  ggsss 

^^1 

o 

SSooo 

»  t-  X  CI  o 

©OOOth 

S2223    2«2'2§ 

S<             2i 

144 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


o 

:  ;*Sg 

sssss 

SgSSEo 

« 

i  :^°.". 

g^s^s 

gS«OOW 

2    1     •  ■  ■  ■  ■ 
'^1     ;  :  ;  ;  : 

;Oggg 

gss^^g 

SSgSoSS 

®gggg 

^^gs_g 

^^^^^ 

»     1 

ffloot-eoT*t 

03  OS  OS  OS  00 

^^^^^ 

y-t 

;    ;    ;    ;0 

gggss 

ggggs 

00  00  00  00  00 

Si  MMi 

:  :  :*S 

gggss 

(N.-IOSQ0  t- 

00  00  GO  CO  CO 

M         ■  :  :  :  : 

:  ;  i^^ 

£§Sgg 

sssss 

sssss 

S         M  ;  i  : 

.      '©OSt^ 

ClCJCSOiCi 

gsssss 

SSSoSS 

;; 

:  :gSS 

lO  -#  CO  .-1  O 

o>oor-!0»fi 

00  OO  00  00  00 

3SS»§ 

I  I  ; ■                                                                 1 

o         :  :  :  :  : 

:«gSg 

sssss 

00  c-  o  o  -* 

sssss 

1 

«g£gg 

ggsss 

sasss 

ssssg 

1 

1 

ggggs 

gsgss 

feSSSS 

(M.-IOOSOO 
OOOOOO  C-C- 

.« 

a 

O 

1  :  ■  ■© 

ssgss 

ggsss 

OO  00  00  00  00 

xS  t-C-C- 

...... 

k 

1 

:  :  ;«§ 

sgssg 

S^gg&S 

10-*  fO(M  .-( 

CJ  OS  xoo  t- 

X  c-c-c- c- 

«(-( 

.     .     .tH 

1 

1 

;  :  i°S 

gg§8£g 

3SS3S 

oosxc-eo 
xt>  t-c-c- 

g 

:  :®gS 

gggSg 

sssss 

OOGC  OOOoS 

c- C-- c-c-c- 

>     .1-1     *     ' 

R 

i     ;  i  M  i 

:«gSS 

o  Ci  OS  c;  CO 

sssss 

00  00  00  00  t- 

X  t-  o  la  la 
c-c-i>c-c- 

o 

:§ggg 

■*  Cl  T-H  O  CO 

cnocicioo 

ooSooSw 

(M.-ICSOS00 
OOOOOOt-C' 

c-c- t-c-t- 

1        i  i  :  i  i 

"=§£§3 

gsgss 

M00«00« 

00  00  OO  t^ 

c-c-c-c- t- 

^     .      .     .     . 

1 

;     ;     ;     ;0 

§Sg^§ 

SgSooS 

SSSooS 

OOSQOCO  t- 

gesEsg 

!     .     .     .w 

a 

1     ;     .     .C5 

sgggg 

sgsss 

O  OS  oot>o 
X  t>  t-  t-  L~ 

i>c:-c- t-c-. 

o 

:  ;  :og 

sgggs 

gSSsSS 

sssss 

t-t-l  •  t-t- 

c- c-c-c- 1- 

•      -0=!^  t~ 

ggggg 

gssss 

sssss 

00  t-OO'* 

t- t- t>  t- r> 

rt  CO  Ol  — !  o 

c-c-  c-c-c- 

§ 

•      •  ClCSOi 

gssss 

SSSbSSS 

(M  -J  o  m  00 

00X00  C- t- 

sssss 

^g2p!g^ 

CS 

>g£S 

gggss 

sssss 

i-(oesoot- 
oooct-t-c- 

t- t-t-c-t- 

Sci^gg 

o 

ogggg 

gggss 

ssssss 

.-iClOOt-O 
00  c- 1-  t>  t- 

c-^-^-l^-t- 

f^^ggg 

i 

ggggg 

ggssss 

sssss 

oooo  t-o 

00  t-  t-  t-  L- 

C- t-t- t-t- 

OOfflOOt- 
t-C~COCOCC 

1 

g 

gggss 

oSoomS 

sssss 

cioot^coin 

Tl'roiM'-'O 
c-c-c- t-t- 

ogggg 

1 

s 

sssgg 

ggass 

SS2SSg 

t-  t-  L-  C-  t- 

SSSBS 

1 

■5r| 

s 

^o 

Sooo» 

§S§§2 

1-H 

ttr-ODCSO 

ssss^ 

at^xeo 

ALTITUDE  TABLES. 


145 


Table  VI. — Differences  of  aUitude  to  tlie  nearest  foot  for  angles  from  1  minute  to  ^  degrees  and  for  distances 

under  1  mile — Continued. 


Angle  of 
elevation. 

Diiference.s  of  elevation  in  feet. 

c     , 

121 

122 

123 

124 

125 

126 

127 

128 

129 

130 

131 

132 

133 

134 

135 

136 

137 

138 

1°  01' 
02 
03 
04 

1°  05' 

06 
07 
08 
09 
1°  10' 

11 
12 
13 
14 
1°  15' 

IG 
17 
IS 
19 
1°  20' 

21 

23 
24 

1°  25' 

26 

27 

28 

29 

1°  30' 



" 

1.0 

.98 

.97 
.96 
.95 
.94 
.93 

.92 
.91 
.90 
.88 
.87 

.99 

.98 
.97 
.96 
.95 
.93 

.92 
.91 
.90 
.89 
.88 

1.0 

.99 
.98 
.96 
.95 
.94 

.93 
.92 
.91 
.90 
.89 

1.0 

.98 
.97 
.96 
.95 

.94 
.93 
.92 
.91 
.90 

.99 
.98 
.97 
.96 

.95 
.94 
.92 
.91 
.9Cf 

1.0 

.99 
.98 
.96 

.95 
.94 
.93 
.92 
.91 

1.  0 

.98 
.97 

.96 
.95 
.94 
.93 
.92 

.99 
.98 

.97 
.96 
.95 
.94 
.93 

1.0 

.99 

.98 
.97 
.95 
.94 
.93 

1.0 

.98 
.97 
.96 
.95 
.94 

.99 
.98 
.97 
.96 
.95 

1.0 

.99 
.98 
.97 
.95 

1.0 

.98 
.97 
.96 

.99 
.98 
.97 

1.0 

.99 
.98 

.99 
.98 

1.0 

.99 

i'.o' 

MON   XXII- 


-10 


146 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


ALTITUDK  TABLES. 


147 


1 

a 

§ 

1 

=4- 

S 

s 

^.^.^^^  ^^^ss  g^^ss  ^^s^^  ^^^^^  s^s^^ 

o 

INWiH^O        OOOaOSOO        QOQOOC-t-        O  !0  CO  iftiD        WdH -*  rj*  M        CO  CO  CO  (M  ca 
-*  Tli -.H -^H -^        Tt1-*MCOCO        COCOCOCOCO        CO  CO  M  CO  CO        CO  CO  CO  CO  CO        COCOCOCOCO 

GO                 ^^^^^         mSmMTO         MMMMTO         OT  03  03  M  M         OT  S  CO  CO  CO         M  ffi  OT  M  CO 

S 

5J§5§g      mSmrom      raSSSm      roeoSSeo      eoSmrofo      mmroraS 

g 

^^^^^  s^^^^  ^^^^^  ^^^^^  ^^^^^  sssss 

^ 

^s_^^^  ^^^^^■^n^^^^^  s^sss  ^^^^^  ^^^^^ 

»0 

^^^^^  ^^^^^  ^s^^^  ^^^^^  ^^^s^  ^^^^^ 

SI 

^^^^^  ^^^^^  s^sss  ss^ss  sssss  S8s_§is 

s 

i>t-eocoto      lo  in  in -^H -Tjf      ■^cococow      cq  w  i-h  .-( i-i      i-(oooo      osoioioooo 

s 

(x>(D«Mnin      in^ttiTjiM      co  co  e<nM  w      w  ^  .-i  o  o      o  o  ci  o  oi      os  oo  oo  oc  oo 

COCOCOCOCO         COCOCOCOCO         COmCOfOCO         COCOCOCOCO         COCOWNSM         (MMtMcaM 

o 

coinmm-*      ^ti*  ••*  eo  co  co      (MMiMiHi-i      «oooo      eiosoiCico      oooocoix:^ 

s 

lO  in  "^  Tff  Tj<         COCONIMW         WrHr-IWO         O  O  OS  Oi  O)         Oi  CO  O)  00  CO         t- c- t- t- t- 
COeOCOfOCO        COCOCOCOCO        COCOCOeOCO        COCOlMtMiM        (MCMCHMiM        M(N(MW(M 

■* 

^SSS35S    S3?3g??S    SSSSS    gSSSS    SSSSS    SSSSS 

rtlCOCOC^CM        (MT-(iHtH.->        OOOOJO)        Ol  Oj  00  00  CO        t- t- t- r-   C~        to  (D  (O  dO  tO 

» 

cocoiN(MC3      i-f-— (OOo      o  Os  Oi  OS  00      00  00  00  t— t—      ir- c- 50  eo  CO      totommin 
OTCOCOCOCO        COCOCOCOCO        CO(M<M(MW        CJMlMfMW        (MW(M<NiM        (M(NC4(MN 

■* 

■* 
■* 

SSSSS    SgSSS    SSSSS    SSSS^    SSSS8    s^s^^ 

^ 

sssss  gsssss  sasss  §§sss  sssssa  aaa^s 

■* 

SSSgS    SSSSSS    SSSSS    SSSSa    ^S^SS    S_?3SS3S3 

•* 

SS§g?S    SgSSS    SSSSS    SSSSiSS    ?S3^S5S3    SSSSSS 

-* 

^s^^^  ^^^^^  ^^^^^  ssas^  ^^^^^  ^^^^^ 

§ 

§§sSSSS    SSSSS    iSSSSSI    S^aSS    E3S3?5S?3    sss^a 

Si 

sssss  sssss  ssaas  ssss^ss  gg?3S3?3  sssas 

« 

SSSSS    iSSSSS    aSSS5?5    g5SS3S3S    SSSSS    SSSSS 

« 

sssss  as^^s  sssss  sss3§3S  assss  sssss 

s? 

^SS^S    SasSSS    l3Sg3?3?3    I3SSSE3    s§§§§    s^^ss 

■* 

SS5SSS    SS5S5g!?3    SS3g3SS    SSSSS    SSSSS    3SSSS 

SI 

SS3S5S3?3    SS?3SS    SSSSS    SSSSS    SSSSS    SSSS3 

S3 

SSS^S.    S3?3SSS    SSSSS    SSSSS    SSS2S    SSSSS 

S 

?qcqc4?qw     wwww^     wSmSS     SSS22      S2SS2     wS^^^ 

o 

148 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


1 

.a 

a 
■J 
> 

i 

s 

i 

5 

O 

C3 

sssss 

sssgg 

gssgg 

gssss 

sssss 

Sgg33 

1 

g 

sssgs 

sgggg 

BBSSg 

gssss 

ggggg 

OOOJOiOO 

o  in  ■*  th  Tj< 

1 

OD 

ggggg 

SSSSK 

ssggg 

^sssss 

(MtHiHOO 

OlOiCJOOOO 
-* -^  Tt< -<*l  tH 

1 

oS^fflS 

gggSB 

gggS3 

SggSS 

SSSS^ 

33SS5; 

1 

» 

ggsss 

ssssg 

gggss 

OOOClOl 

5355:5; 

1 

s 

ggggS 

i-T  if3  in  o  in 

gsssg 

ITS  iTi  in  vfi  irs 

00  0:05  00 

lO  -«'  -#  Tjl  ■* 

3S55S 

1 

-1 

OgSiOOOO 

Kgggg 

sssgg 

gssss 

SSS3S 

t-t-CO  tO!S 

ggSBE 

ggsss 

SSggo 

Sggg" 

§^35:!5 

5S§!S!5 

s 

giSBKS 

SSSSS 

ggggs 

ggsss 

OOOOC-t-tO 

^^!g!g5; 

s 

ggggg 

sasss 

ggssg 

gS§32S 

Qor-t>»  CO 

igS!535 

® 

KSSSS 

sssgg 

ca— '.-Hoo 

ClOCOQO  t- 

55;SSS 

!S3gi3;3 

l> 

ggsss 

ssggg 

m  in  m  **  -f. 

53^55; 

SSSSiS 

^3:33133 

s 

gggss 

ssssa 

in  in-*  ^is' 

00t>  C-t-tO 

s;gS55 

S333§ 

l> 

sssss 

ggssg 

OCSC5  O)  00 

^^^^5 

!g§3i5!3 

S3S3S 

sssss 

gssgg 

§§§5;^ 

5^^^^ 

i§S3^S 

3333^    1 

t* 

sssss 

Sgg^S 

^^^^^ 

CO  'O  O  ift  -* 

331553 

3335:5! 

s 

Si^SSS 

gssss 

^^^^i? 

!SS§53 

333^3 

3.... 

IT- 

ggSog 

o  c:  Ci  00  00 

lO  rj(  T# -rjl  ^ 

t-  t-  O  '-S  US 

^^^^" 

-^   -*  Tjl  -^  T}< 

rHrHOO  0 

c* 

.-1^  ooei 

§1?«S:5: 

3I3J35S 

■*  -rP  -*  Tj<  -^Jl 

0  0  OCl  Oi 

s 

SSg3S 

51S5^S 

la  u'5  m  -*  ^ 

:SS39S 

0  omcios 

^  T1-05C0C0 

o 

g?§?5 

!gs3i3g 

""^^^ 

553SS 

sssss 

« 

SSS3!; 

t-eoeoiairs 

■*  "<ji  ■*  1*  ■* 

:3;:S^3S 

^3^5!  5! 

OOOCl  Oi 

gssss 

9 

^^55^ 

55-33SS 

g=!55!S 

O  OOJOJOl 
■Til  -*  cc  CO  CO 

gSSSKR 

r- 

o 

S555S 

s^s^s 

33^5!^ 

51515SS 

gggJSS 

o?McoMeo 

«£ 
« 

Jj^^^!^ 

353I3S 

ggggjg! 

.-I  O  Offi  O) 
-fTjl  -^CO  CO 

gssss 

sssss 

§ 

^SSig^ 

53:^93 

N^fH  .-I  O 

O  OOiOlOO 

-*  coco  MCO 

CO  t^c-c-r- 

sssss 

■* 
9 

s^igs^ 

5359^ 

5!5S§^ 

gsgsg 

ooc-t-ixo 

sgssss 

» 

-a. -* -tf -fj- Tjl 

39^:55 

SSSSS 

sssss 

KSSgg 

SSISSS 

« 

3^sgg 

<M«i-HrHO 

OOOSOSCO 

mSRmra 

ggSgi? 

i?S3SSg 

9 

^33^^ 

,-H^.-IO  O 

gg^^fg^ 

sssss 

sssss 

sssss 

■3>|.2 

o 

|ii?;s| 

5?:55^ 

sssss 

«i>.XCSO 

ALTITUDE  TABLES. 


149 


> 
■1 

i 

£ 
P 

s 

sssss 

SSSSg 

COt-OOiO 
t-  l>  C-  L-  C- 

C-C-D-t-t- 

t-4o  oosai 

oot-r-CD  to 

toco  to  CD  to 



o 

■-■ooioor- 

O0  00l>l>  t- 

CO  CI  (M  tH  0 

t-c-e-t-t- 

Sgggo 

Sgggg 

1-H 

mSoqooooo 

COt- t>  t-I> 

t-c-c- t-t- 

OTM  —  00 
C'  t-  t-  t^  t^ 

gggss 

toco  to  toco 



1H 

3S3SS§ 

010)00  i>o 

t- t-t-t-t- 

gSfi^gg 

ggggg 

gggss 



o 

OlCOt^OlO 
O  C-OC-t- 

SP:f3Kg 

f:pi^§S 

S££Sg 

ggggs 

iS 

gssssg 

00  i>  O  to  lO 

t-i>i>c-r> 

T-lO©QOCO 

D- t>  tOtO«3 

BSggg 

SSSgg 



■* 

(M.-I00100 
oocoooc- t- 

t>r>c-i>  t- 

ggg§£ 

ogggg 

SSSSg 

iH 

.-HOOlCOt- 

oooot-c-t- 

t- to  in  'St  CO 

t-t-l?-  t~c- 

gggss 

ggS3S 

ggggg 

IH 

00  t- t-t- t- 

tDiO-»#TiicO 

t-c-t-t- 1- 

WW  rHOC» 

t-t-c-t- to 

gg£SS 

gggSg 

toeoooeo 

iH 

gSScSc^S 

lfi»OTHcO(M 

t-t- t- ^>t:- 

w  wo  OSOJ 

t^  t-  t^  CD  to 

ggogg 

ggsss 

SSSgg 

O 

i>c>D- ^-t- 

t-  i>  c- 1-  t> 

wo  OOJOO 
t-t-t-CDCD 

ggggg 

SSSSS 

SSSgg 

i 

I^•  t-t- t- t- 

OOCSOOOO 
t-l>tD  tOCD 

£§S©S 

sssss 

SSSgg 

© 

t- to  CO  to-* 
t-c-i>  t-t- 

coco  tM  wo 
t- t>  t-C-t- 

OOi  00  00  t- 
t>  CD  to  CD  CO 

gggg^ 

COCOCVJlM  w 

to  CO  to  CO  to 

Sgggg 

© 

t^t- t~  t-c- 

eOCQr-iOO 

t-t- t>t-l> 

ggsss 

ggsss 

gggss 

ggggg 

© 

c-t-r-c-t- 

(M-HOOOl 

c-t*i>t- to 

CD  to  to  to  CD 

gssss 

(M(M  wo  0 

gggss 

us 

O 

sssss 

i>t-c-to  to 

to  to  toco  to 

gssss 

(M  WOOOl 

COCO  CD  too 

gggss 

S 

t>c- t-^  t- t- 

Edgggg 

tOCD  to  CD  to 

SSggg 

sgggg 

11 

^S?2S^ 

tr-to  to  toto 

CD  CD  to  CD  CD 

CO?^(M(M  W 
to  CO  CD  CD  to 

gggss 

00t>r-CD(O 

Tl 

«(M  WiHO 
t>C-t-l>t> 

ggSBg 

gggss 

gggsg 

ggggs 

SSggg 

© 
1-( 

c-c-c- t-to 

ggggg 

sssss 

IN  W  woo 

ggSSB 

sgggg 

g2E:!gg8 

to  to  to  to  to 

S3SSS 

ssggg 

gSSBS 

ggggs 

© 

t- t-CD  to  to 

to  to  to  to  to 

SSSSS 

woo  cri  01 

gSBgg 

gg3S3 

© 

gggss 

sggss 

SSSiSsS 

SSSSS 

KBggg 

sssgg 

© 

SSosoco 

Sg3SS 

SS3Sg 

OOCOOOt- 

tDioo  mm 

EgggiS 

ioiSggg 

© 
© 

SoStoto 

S3SSS 

ssssg 

CJOOCO  t>l~- 

in  tnmmm 

gggss 

ssggg  1 

§ 

S£SSS 

CD  to  to  to  to 

ssgss 

SSKBS 

iftmiomm 

coco  !MiM  W 

irsmmioo 

© 

SSSgg 

ssggg 

sgggg 

SKSSS 

mlnmmS? 

Smomm       1 

© 

sgsgss 

8SOSJ0 

ssggs? 

KKSSS 

m-^-tfcoco 
lommirso 

SS^^Sg      1 

© 

sssss 

(MfMi-HO  0 
CO  to  to  to  CO 

SSSSK 

Kggsa 

;sss?sg 

WW woo       1 
ift  10  m  m  in 

© 

S33SS 

(M  wo  001 

SSSKg 

sssss 

COCOMfMCM 

inioiniriio 

ssgg^   1 

° 

T-(                       IH 

SSi^l 

■*-*'*■*•* 

iH 

©c-co©© 

150 


A  MANUAL  OF  TOP0GEA.PHIC  METHODS. 


a 

a 

1 
I 

1 
1 

o 

:  i'.^". 

sssss 

SSSSoo 

g^SS^ 

SSSSoSS 

i 

1 

i'ggS 

IlH      '      '      * 

g§sg§ 

CMi-HOCJOO 

ojcnoiooco 

goSoS^S 

SSSSffiS 

5 

•* 

iggss 

sssgg 

ggssss 

SSSS^^ 

ssssss 

^ 

t» 

^    '    *    '    ■ 

gssgg 

^^-M 

Smodotoo 

«i(M.-(00 

= 

« 

•* 

^ 

■    ;    ;    ;« 

oioot- to»n 

ssssg 

ggssss 

00  QO  00  00  00 

eg  5^0001 

"^ 

:::;'' 

3 

s 

;  ;  i  is 

ojoj  oicncs 

gssss 

ssgass 

ift-^JtCOCOW 

r-li-HOOsm 

00  00  oot-t- 

S 

:  :  i  ! 1 

£ 

^ 
■* 

ggggs 

M  «  -H  O  OS 

ojoiojoaoo 

ssssss 

sssss 

r-iO  0105  00 

oooot-c-t- 

3 

s 

;  :*o?m 

sssss 

OJOSOIOOOO 

8SSSSS 

sssss 

oooiooco 

00  00  t-  t-  L-- 

^ 

1-1 

•      •  O500  t- 

SraSSro 

sssssg 

sssss 

COCO(Mi-HO 

ooi>  t^t>  r- 

»-» 

g 

-* 

:«ggS 

SSSSo 

SSotoooo 

sssss 

sgssss 

C-t-  I>  t-t- 

1 

o 

'ggSS 

gsggs 

o  csOi  O)  tr- 

SffiSSoo 

w^rHom 

ooo  [>  too 

^ 

§ 

% 

gssgg 

sggss 

oooot-co 

OS  CO  CO  00  00 

ssssss 

g353SgS 

l>t>t>  t-t- 

3<; 

S 

ggfegg 

gsgss 

§g£§S 

sssss 

i-HOClCi  00 

oooot- t>t> 

r^D^Sb^t^ 

S 

g 

SSSSS 

ggj^gg 

ggssss 

00  00  00  00  CO 

0005COOO 
COQOI>I>C- 

l>(0  tOlO  tH 

C-  I-  [>  t-  t-        1 

«s 

SSSSS 

gssss 

sssss 

OTOOOOOoS 

QDt>l>t- t- 

t-c-t>  t-t-      1 

1 

1-1 

SSSSS 

gs§§s 

O0»S^» 

ssssg 

t-  I>  l^  c-  t- 

i>r>  t-c-t- 

■^ 

ir- t-^>c-t> 

§ 

-* 

sssss 

ggsss 

SwSSqo 

ssss?; 

l?-t- t- t-t- 

& 

e 

iH 

gsggg 

OTOTMOOaO 

sssss 

sssse 

tr- t- t- t-C- 

t- t- t-c-t- 

•- 

< 

sssgs 

O0iC0t>!0 

igSS3S3g 

ssseg 

t-Ot-D-D- 

l>t>l>I>D- 

O 

< 

sggss 

ssssbss 

SSS5SS 

.-(Oo;co  t- 
OOODC- t-C- 

t- t>  t- t-t- 

s 

© 

ggggg 

00  t-^^g 

SoDTOOTOO 

OOiOOOOt- 
OOt-t-C't- 

CD  ID  U3  -tjt  -* 

t>i>  t>c-P      1 

» 

a> 

i        C0.-i00i00 

1      oiOiojcooo 

OOOO^OOM 

SSSsSS 

?i"c2t^S 

SSSgg 

^ 

5 

s 

gsgss 

sssss 

(71(M  .-H  O  Oi 
00  00  00  00  t- 

t- t-t-t-D- 

t-D-t-C-t- 

ssggs 

t» 

SgggBSS 

SSSoOOD 

(Mi-HOO  C3 
00  CO  00  00  t- 

tS^t^l^t^ 

t-ot- t- t* 

D^t- t-CD  to        1 

"-« 

°«Sqo» 

sssss 

OOOOOOt- t- 

cooeoiom 
i>c-i>  t-c- 

t>t^c-c*t- 

Sgggg 

i 

ssssss 

gssss 

00  00  c-  t-c- 

05  «  N  t-H  r-l 

gggss 

a 

C30O0O0  00CO 

SS5S5S53 

ssggs 

c-c-t-t- t- 

COlMr-ii-tO 
t-C-t- t-C- 

essss 

^ 

s 

SoooOTOom 

sssss 

Oiooot' to 

t^  C~  l>  t-  D- 

iT]  M  .-H  O  O 

sssss 

1 

> 

s 

sss^s 

sssse 

oiQOtxoeo 

D-D-C-t-t- 

(M.H0001 

c-t-t-c-to 

ggSSB 

s 

s 

SSSSS3 

OOOO  COCOC- 

t-  t-  c- 1-  t- 

F!^°§§ 

SSBSS 

H 

5^1 

o 

iH  CI  M  -*  O 

sssss 

eaD-coosb 

ALTITUDE  TABLES. 


151 


1 

o.S 

1 
s 

-* 

■  -Z 

1 

;     i  iggg 

1 

,  i*.^ 

1H 

;      ;«ggS 

1 

iss 

t> 

;     ©gggg; 

;    .H  '  ■  ■  ■ 

©gg 

l> 

;    SSSSS 

°        ft 

*  ggggg 

;  ;  I'g   gsggg   1 

-*       1         :^!oi3j      QOi-:oin-* 

eo      1        :    jog^      ^^gg^ 

S       ;®g§s  ssssg 

s   1    jgggS   gssgg 

^      1     ^ 

1 

a 
a    . 

n 

O 

;  ;  :*  g»£gs  ssssss 

<» 

i  ;  jg    ggSSS    3Sgg3 

c* 

o 

i  i»g  ssssg  ggsss 

1 

'i-( 

» 

iSg^  sssgg  sgsss 

■* 

;oggs  sgsss   gsssg 

fH 

SIOJOIOJ        OiOOSOSOi        CIOIOJQOCO 

w 

Og^5;g       mosSSOT       SSSwoo 

?0 

*    gg&gg    SSSgg    SSSSSES 

1 

©OS        Q0lr~CDCOurS        -r^tcOMiH-^        OOiOOC-t^ 

s 

Sg    g£gS3    SgSSS    SSSESS 

us 

©moo      t:toi23'S2      co  m  rH  o  o>      c»  oo  t- to  as 
^ci  o>      oiOTOJC;  OS      ojoioJOToo      oooooooooo 

c» 

sss  ssssg  sssss-ssass 

o 

®g^s  g_^s^gg  ^ggsss  ggssss 

3 

.  ^^^^^  s^^ss^  ^_^_^^^  ^sss^  | 

■  ■©        CSQOt^COin        in^MMr-l        OOOiOOC~        lO^lOHH-* 

■  ■"        a>OOJ0505        OSCJOOSOl        OiCiOOMOO        ODCOOOOOOO 

U 

■      .  0>        O00OC~tDlft        'itcCCq^O        OOJOOt>t-        tDO^-tCO 

!S    1     ■  ; 

:»g  ssssg  ggssg  gggass  ggsss 

ifS 

•  OiOi       mojoiciOJ      oicnojcioo      oooooooooo       oooooooooo 

«  i  fl 

o 

i4«»-*us      «0f  xcs©      rtMM-H-io      ©i>-ao©o      — ei«-*tffl      ©t^QO©© 
MWWMM       «»»03-*       M''*-*'*-*       •^■*-*'*ia       lOUSiOUSiS       »0»«i«»i5© 

152  A  MANUAL  OF  TOPOGKAPHIC  METHODS. 

Table  VII. — Differences  of  altitude  from  angles  of  elevation  or  depression. 


.  ,^..    J         c  +  DA,  +  fti  for  angles  of  elevation. 

Difference  of  altitude  =  \lj)k\  +  ft^  for  angles  of  depression. 

D= distance 
ft,  =  5280  ft. 
refraction. 

m miles,  a  =  angle  of  elevation  or  depression ; 
•<  tan  a;  h^  :^  correction  for  curvature  and 
A-rgument  for  ft,  is  a';  argument  for  A2  is  ^• 

0° 

1° 

2° 

3° 

40 

6° 

6° 

JO 

8° 

90 

10° 

11° 

.  12° 

13° 

14° 

16° 

hi 

hi 

l>j 

hi 

hi 

hi 

hi 

hi 

hi 

hi 

Feet. 

h. 

h. 

hi 

Feet. 

hi 

hi 

hi 

, 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.0 

92.2 

184.4 

276.7 

369  2 

461.9 

555.0 

648.3 

742.0 

836.3 

931.0 

1026. 3 

1122.  3 

1219.  0 

1316.  5 

1414.8 

1 

1.5 

93.7 

185.9 

278.2 

370.7 

463.5 

556.5 

649  9 

743.6 

837.8 

932.6 

1027.  9 

1123.  9 

1220.  6 

1318, 1 

1416. 4 

■2 

3.1 

95.2 

187.4 

279  8 

372.3 

465.0 

558.0 

651.4 

745.2 

839.  4 

934.2 

1029. 5 

1125.5 

1222. 2 

1319,7;  1418,  0| 

3 

4.6 

96.8 

189.0 

281.3 

373.8 

466.6 

559.6 

653.  0 

746.7 

841.0 

935.  8 

1031.1 

1127.1 

1223. 8 

1321,  3 

1419.  7 

4 

6.1 

98.3 

190.5 

282.9 

375.4 

468.1 

561.2 

654.5 

748.3 

842.6 

937.3 

1032.7 

1128.  7 

1225.  5 

1323. 0 

1421.3 

6 

7.7 

99.8 

192.1 

284.4 

376.9 

469.7 

562.7 

656.1 

749  9 

844.1 

938.  9 

1034.  3 

1130.  3 

1227. 1 

1324.  6 

1423.  0 

6 

9.2 

101.4 

193.6 

286.0 

378.5 

471.2 

564.3 

657.7 

731.4 

845.7 

940.5 

1035, 9 

1131.9 

1228.  7 

1326.  2 

1424.  6 

10.7 

102.9 

195.1 

287.  5 

380.0 

472.8 

565.8 

659  2 

753.0 

847.3 

942.1 

1037.  5 

1133.  5 

1230.  3 

1327.  9 

1423  -9 

8 

12.3 

104.4 

196.7 

289  0 

381.6 

474.3 

567.4 

660.8 

754.6 

848.9 

943.7 

1039.  1 

1135. 2 

1331.9 

1329,  5 

1427.  9 

9 

13.8 

106.0 

198.2 

290.6 

383.1 

475.9 

568.9 

662.3 

756.1 

850.4 

945.3 

1040.7 

1136.  8 

1233.  6 

1331,1 

1429.  6 

10 

15.4 

107.5 

199. 8     292. 1 

384.6 

477.4 

570.5 

663.9 

757.7 

852.0 

946.8 

1042.  3 

1138.  4 

1235.  2 

1332.  8 

1431. 2 

11 

16.9 

109.1 

201.3 

293.  7 

386.2 

479  0 

572.0 

665.5 

759  3 

853.6 

948.4 

1043.  8 

1140.  0 

1236,  8 

1334,4 

1432.9 

12 

18.4 

110.  6 

202.8 

295.2 

387.7 

480.5 

573.6 

667.0 

760.9 

855.2 

950.0 

1045.  4 

1141.  6 

1238.4 

1335,0    1434.5 

13 

20.0 

112.1 

204.  4 

296.7 

389  3 

482.1 

575.1 

668.6 

762.4 

856.8 

951.  0 

1047.0 

1143.2    1240.0 

1337,7    1436.2 

14 

21.5 

113.7 

205.  91    298.  3 

390.8 

483.6 

576.7 

670.1 

705.0 

858.3 

953.2 

1048.  6 

1144.8    1241.7 

1339,3    1437.8 

15 

23.0 

115.2 

207.  5!     299. 8 

392.4 

485.2 

578.2 

671.7 

765.6 

859.9 

954.  7 

1050.  2 

1146.  4|  1243,3 

1340.9    1439  5 

16 

24.6 

116.7 

209.0;    301.3 

393.9 

486.7 

579.8 

673.3 

767.1 

861.5 

956.3:  1051.8 

1148,0;  1244.9 

1342.6    1441.1 

17 

26.1    118.3 

210.  5     302. 9 

395.5 

488.3 

581.3 

674.8 

768.7 

863.0 

9,57,9    1053.4 

1149.6.  1246,5 

1344.2    1442.8 

IS 

27.6    119.8 

212. 1     304. 4 

397.0 

489.8 

582.9 

676.4 

770.3 

864.6 

959  5   1055.0 

1151.  2|  1248.1 

1345.8,  1444.4 

19 

29.2 

121.4 

213. 6      308.  0 

398.6 

491.3 

584.4 

077.9 

771.8 

866.2 

961.1    1056.6 

1152.8 

1249,  8 

1347.5;  1446.1 

■20 

30.7 

122.9 

215. 1,     307.  5 

400.1 

492.9 

586.0 

679  5 

773.4 

867.8 

962.7    1058.2 

1154.4 

1251,  4 

13491,  1447.7 

21 

32.3 

124.4 

216. 7      309.  1 

401.6 

494.5 

587.6 

681.1 

775.0 

869  4 

964.3    1059.8 

1156. 1 

1253,  0 

1350.  8^   1449  4 

22 

33.8 

126.0 

218.2     310.6 

403.2 

496.0 

5891 

682.6 

776.5 

870.0 

965,  9,  1061.  4 

1157.7 

1254.  6 

1352.4    1451.0 

23 

35.3 

127.5 

219  8|     312.1 

404.7 

497.6 

590.7 

684.2 

778.1 

872.5 

967.51  1063,0 

1159.3 

1256.  2 

1354.0    1452.7 

24 

36.9 

129.0 

221.  3      313.  7 

406.3 

499.1 

592.  2 

685.7 

779  7 

874.1 

969  0    1064.6'  1100.  9|  12.'>7.  9 

1355.7,  1454.4 

25 

38.4 

130.6 

222.8 

315.2 

407.8 

50O7 

593.8 

687.3 

781.3 

875.7 

970.  6  1066.  2 

1162.  5|  1259.5 

1357.3    1456.0 

26 

39.9 

132.1 

224.4 

316.8 

409.4 

502.2 

595.4 

688.9 

782.8 

877.3 

972.2    1067.8 

1164. 1:  1261. 1 

1358.9    1457.7 

27 

41.5 

133.6    225.9 

318.3 

410.9 

503.8 

596.9 

690.4 

784.4 

878,8 

973.8    1069  4 

1165,7)  1162,7 

1360.6    1459  3 

28 

43.0 

135.2   227.4 

319  9 

412.5 

505.3 

598.5 

692.0 

786.0 

880  4 

975.  4I  1071.  0 

1167.  3 

1264.  4 

1362.2    1461.0 

29 

44.5 

136.7    229.0 

321.4 

414.0 

506.9 

600.0 

693.6 

787.5 

882.  0 

977,  0'  1072,  6 

1168.  9 

1266.  0 

1363.9,  1462.6 

30 

46.1 

138.3 

230.  5 

322.9 

415.5 

508.4 

601.6 

695.1 

7891 

883.6 

978,  6.  1074.  2 

1170.  6 

1267.  6 

1365.5!  1464.3 

31 

47.6 

139  8 

232.1 

324.5 

417.1 

510.0 

603.1 

696.7 

790.7 

885.1 

980.1'  1075.8 

1172.2 

1269.  3 

1367,1!  1465.9 

32 

49.2 

141.3 

233.6 

326.0 

418.6 

511.5 

604.7 

698.2 

792.2 

886.7 

981.  7,  1077.  4 

1173. 8 

1270,9 

1368.8   1467.6 

33 

50.7 

142.9 

235.1 

327.6 

420.2 

513.0 

606.2 

699  8 

793.8 

888.3 

983,3    1079,0 

1175. 4 

1272,  5 

1370.4    1469  2 

34 

52.2 

144.4 

236.7 

329.1 

421.7 

514.6 

607.8 

701.4 

795.4 

889  9 

984,9    108O6 

1177.  0 

1274,  1 

1372. 1 

1470.  9 

35 

53.8 

146.0 

238.2 

330.6 

423.3 

516.2 

601.3 

702.9 

797.0 

891.5 

986.5   1082.2 

1178.  6 

1275.  7 

1373.  7 

1472.  5 

36 

55.3 

147.5 

239  8 

332.2 

424.8 

517.7 

610.9 

704.5 

798.5 

893.  ( 

988.  li  1083.8 

1180.  2 

1277.  4 

1375.  3 

1474,  2 

37 

56.8 

149.0 

241.3 

333.7 

426.4 

519.3 

612.5 

706.1 

800.1 

894.6 

989.7 

1085. 4 

1181.8 

1279,  0 

1377.  0 

1475.  9 

38 

58.4 

150.6 

242.8 

335.3 

427.9 

520.8 

614.  0     707.  6 

801.7 

896.2 

991.3 

1087.  0 

1183.4 

1280. 6 

1378.  6 

1477.  5 

39 

59.9 

152.1 

244.4 

336.8 

429  5 

522.4 

615.6      709.2 

803.2 

897.8 

992.9 

1088.  6 

1185.  0 

1282. 2 

1.380.  3 

1479.2 

40 

61.4 

153.6 

2J5.9 

338.4 

431.0 

523.9 

617.1     710.7 

804.8 

899  4 

994.5 

1090.  21  1186.  71  1283.  9 

1381.  9 

1480.  8 

41 

63.0 

155.2   247.5 

339  9 

432.6 

525.5 

618. 7     712.  3 

806.4 

900.9 

990,0 

1091.81  1188.3!  1285.5 

1383.  5 

1482.  5 

42 

64.5 

156.7   249  0 

341.4 

i     434.1 

527.0 

620.21    713.9 

807.9 

902.5 

997.6 

1093.4 

1189.  9 

1287. 1 

1385.2 

1484. 1 

43 

66.0 

158.2   250.5 

343.0 

1    435.6 

528.6 

621.8 

715.4 

809.5 

904.1 

999  2 

1095.  0 

1191.  5 

1288.  8 

1386.  8 

1485.  8 

44 

67.6 

159  8   252.1 

344.0 

'    437.2 

530.1 

623.3 

717.0 

811.1 

905.7 

1000.  8 

1096.  6 

1193. 1 

1290.  4 

1388.  6 

1487.  5 

45 

69.1 

161.3    253.6 

346.1 

438.7 

531.7 

624.9 

718.6 

812.7 

907.3 

1002.4 

1098.  2 

1194.  7 

1292.  0 

1390. 1 

1489. 1 

46 

70.6 

162.9  255.1 

347.  e 

440.3 

533.2 

626.4 

720.1 

814.2 

908.8 

1004.  0 

1099,  8 

1196.  3 

1293.  7 

1391.  8 

1490. 8 

47 

72.2 

164.4   256.'- 

3491 

441.8 

534.8 

628.0 

721.7 

815.8 

910  4 

1005.  6 

1101.  r 

1197.  9 

1295.  3 

1393.  4 

1492.  4 

48 

73.7 

165.9   258.2 

350.- 

443.4 

536.3 

629  6 

723.3 

817,4 

912,0 

1007.  2 

1103. 1 

1199.  6 

1296.  9 

1395. 0 

1494. 1 

49 

75.3 

167.5    259.  S 

352.2 

444.9 

537.9 

631.1 

724.8 

819  0 

913.6 

:  1008.8 

1104. 7 

1201.  2 

1298.  5 

1396.  7 

1495.8 

50 

76.  S 

169.0    261.3 

353.  S 

446.5 

539.4 

632.7 

726.4 

820.5 

915.2 

1010,  4 

1106.  3 

1202.  8 

1300.  2 

1398.3 

1497.4 

51 

78.1 

170.6   262.  f 

355. 

448.0 

541.0 

634.2 

728.0 

822.1 

916.' 

1012,  0 

1107.  9 

1204. 4 

1301.  8 

1400,  0 

1499. 1 

52 

79.  t 

172.1;  264.4 

356. 

449.6 

542.5 

635.  f 

729  5 

823.7 

918,? 

1013,  6 

1109,  5 

1206,  0 

1303,4 

1401.6 

1500. 7 

S3 

81.4 

173.6    265.  £ 

358..: 

451.1 

544.1 

637.  L 

731.1 

825.2 

919.  £ 

1015.  2 

1111,1 

1207. ' 

1305.0 

1403,  3 

1502,4 

54 

82.  < 

176.2    267.. 

360. 

452.' 

545.1 

638.  £ 

732.7 

826.8 

921.  c 

1016.  i 

1112.' 

1209  3 

1306.7 

1404.9,  1504.1 

55 

84.. 

176. 7 1  269  ( 

361. 

454.2 

547.2 

640.  4'     734.  2 

828.4 

923.1 

1018.4 

1114,3 

1210,  S 

1308.  3 

1406.5    1505.7 

56 

86. 

178.  2:  270. 

363. 

)     455.  i 

548.' 

642.  C 

735.8 

830.  C 

924.' 

1020.  C 

1115, £ 

1212,  e 

1309  9 

1408.21  1507.4 

57 

87. 

) .  179.  8    272. 

364. 

3     457. 

550.: 

643. 

737.4 

831.. 

926.' 

1021.  £ 

1117,  £ 

1214, 1 

1311.  6 

1409,8;  1509  0 

58 

89. 

181.3    273.6     366. 

I     458.' 

551.! 

645. 

738.  a 

833. 

927. 

1023. 1 

1119. 

1215, i 

1313.2;  1410.51  1510.7 

59 

90. 

3   182.9   275.2     367. 

7     460. 

553.4 

646. 

740.  E 

834.' 

929' 

1024. 

1120. 

1217.4 

1314.8    1413,1]  1512.4 

60 

92. 

i    184.4   276.7     369. 

2     461. 

555.  ( 

648. 

742.  C 

836. 

931. 

1026. 

1122, 

1219. 

1316.5   1414.8   1514.0 

ALTITUDE  TABLES. 


153 


Table  VIII. — Corrections  for  curvature  and  refraction. 


D 

hz 

0 

ll2 

B 

hj 

D 

h. 

Miles. 

Feet. 

Miles. 

Feet. 

Miles. 

Feet. 

Miles. 

Feet. 

1.0 

0.6 

5.5 

17.3 

1.1 

0.7 

5.6 

18.0 

3.6 

7.4 

8.1 

37.6 

1.2 

0.8 

5.7 

18.6 

3.7 

7.8 

8.2 

38.6 

1.3 

1.0 

5.8 

19.3 

3.8 

8.3 

8.3 

39.5 

1.4 

1.1 

5.9 

20.0 

3.9 

8.7 

8.4 

40.5 

1.5 

1.3 

6.0 

20.6 

4.0 

9.2 

8.5 

41.4 

1.6 

1.5 

6.1 

21.3 

4.1 

9.6 

8.6 

42.4 

1.7 

1.7 

6.2 

22.0 

4.2 

10.1 

8.7 

43.4 

1.8 

1.9 

6.3 

22.8 

4.3 

10.6 

8.8 

44.4 

1.9 

2.1 

6.4 

23.5 

4.4 

11.1 

8.9 

45.4 

2.0 

2.3 

6.5 

24.2 

4.5 

11.6 

9.0 

46.4 

2.1 

2.5 

6.6 

25.0 

4.6 

12.1 

9.1 

47.5 

2.2 

2.8 

6.7 

25.7 

4.7 

12.7 

9.2 

48.5 

2.3 

3.0 

6.8 

20.5   ■ 

4.8 

13.2 

9.3 

49.6 

2.4 

3.3 

6.9 

27.3 

4.9 

13.8 

9.4 

50.7 

2.5 

3.6 

7.0 

28.1 

5.0 

14.3 

9.5 

51.7 

2.6 

3.9 

7.1 

28.9 

5.1 

14.9 

9.6 

52.8. 

2.7 

4.2 

7.2 

29.7 

5.2 

15.5 

9.7 

53.9 

2.8 

4.5 

7.3 

30.5 

5.3 

16.1 

9.8 

55.1 

2.9 

4.8 

7.4 

31.4 

5.4 

16.7 

9.9 

56.2 

3.0 

5.2 

7.5 

32.2 

5.5 

17.3 

10.0 

57.3 

8.1 

5.5 

7.6 

33.1 

3.2 

5.9 

7.7 

34.0 

3.3 

6.2 

7.8 

34.9 

3.4 

6.6 

7.9 

35.8 

3.5 

7.0 

8.0 

36.7 

154 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  IX. — For  computlnri  differences  of  altitude  from  angles  o) 

scale  1:45000). 
[Prepared  by  li.  S.  Wooclwartl.] 
_.„  -  1^..    ,         C  +  D7ii  +  /ia&r  angles  of  elevation. 

Difterenceof  altit-:icl6=  ^  _  j,;,^  ^  /^fgj  angles  of  depression. 


elev:at%on  or  depression  (applicable  to 


T>  —  distance  in  scale  divisions  ^s  inch  each;  a  ■- 
tion  for  curvature  and  refraction. 

Argument,  for  Ai  is  a ;  argument  for  ftj  is  D. 


angle  of  elevation  or  depression;  fti  =  75  feet  X  tan  a;  7i2  =  correc- 


fti  in  feet.                                                              | 

D 

fta 

D 

Ih 

' 

0° 

1° 

20 

3° 

4° 

5" 

6° 

7° 

Scale 
livisions. 

Feet. 

Scale 
iivisions.j 

Feet. 

0 

.000 

1.309 

2.619 

3.931 

5.245 

6.562 

7.882 

9.208 

00 

0 

720 

60 

1 

.022    1 

1.331 

2.641 

3.952 

5.266 

6.583 

7.905 

9. 231 

93 

1 

726 

61 

.043 

1.353 

2.  662 

3.974 

5.288 

6.605 

7.927 

9.253 

131 

2 

732 

62 

3 

.065 

1.375 

2.684 

3.996 

5.310 

6.628 

7.949 

9.275 

161 

3 

738 

63 

1 

.087 

1.396 

2.707 

4.018 

5.332 

6.649 

7.971 

9.298 

1S6 

4 

744 

64 

5 

.109 

1.418 

2.728 

4.040 

5.354 

6.671 

7.993 

9.319 

208 

5 

750 

65 

6 

.131 

1.440 

2.750 

4.062 

5.376 

6.694 

8.015 

9.342 

228 

6 

755 

66 

.153 

1.462 

2.772 

4.084 

•5.  398 

6.715 

8.037 

9.364 

246 

7 

761 

67 

8 

.175 

1.483 

2.794 

4.105 

5.420 

6.737 

8.059 

9.386 

263 

8 

767 

68 

9 

.196 

1.505 

2.815 

4.127 

5.442 

6.760 

8.081 

9.408 

279 

9 

772 

69 

10 

.218 

1.527 

2.837 

4.150 

5.464 

6.781 

8.104 

9.430 

294 

10 

778 

70 

11 

.240 

1.549 

2. 859 

4.171 

5.485 

6.803 

8.125 

9.452 

308 

11 

783 

71 

12 

.262 

1.571 

2.881 

4.193 

5.508 

6.826 

8.147 

9.475 

322 

12 

789 

72 

13 

.283 

1.593 

2.903 

4.215 

5.  530 

6.847 

8.170 

9.496 

335 

13 

794 

73 

14 

.305 

1.615 

2.925 

4.237 

5.551 

6.869 

8.191 

9.519 

348 

14 

800 

74 

15 

.327 

1.636 

2.947   • 

4.258 

5.573 

6.892 

8.214 

9.541 

360 

15 

805 

75 

16 

.349 

1.658 

2.968 

4.281 

5.596 

6.913 

8.236 

9.563 

372 

16 

811 

76 

i; 

.371 

1.680 

2.990 

4.303 

5.617 

6.935 

8.258 

9.586 

383 

17 

816 

77 

18 

.393 

1.702 

3.012 

4.324 

6.639 

6.958 

8.280 

9.607 

394 

18 

821 

78 

10 

.415 

1.723 

3.034 

4.346 

5.661 

6.979 

8.302 

9.630 

405 

19 

826 

79 

so 

.436 

1.746 

3.056 

4.368 

5.683 

7.001 

8.324 

9.652 

416 

20 

832 

80 

21 

.458 

1.768 

3.078 

4.390 

5.705 

7.024 

8.346 

9.674 

426 

21 

837 

81 

22 

.480 

1.789 

3.100 

4.412 

5.727 

7.045 

8.368 

9.697 

436 

22 

842 

82 

23 

.502 

1.811 

3.121 

4.434 

5.749 

7.067 

8.  390 

9.718 

446 

23 

847 

83 

24 

.523 

1.833 

3.143 

4.456 

5.771 

7.090 

8.413 

9.741 

455 

24 

852 

84 

25 

.545 

1.855 

3.165 

4.477 

5.793 

7.111 

8.434 

9.763 

465 

25 

857 

85 

26 

.567 

1.875 

3.187 

4.499 

5.815 

7.133 

8.457 

9.785 

474 

26 

862 

86 

27 

.589 

1.898 

3.209 

4.522 

5.836 

7.156 

8.479 

9.807 

483 

27 

867 

87 

28 

.610 

1.920 

3.231 

4.543 

5.  859 

7.177 

8.501 

9.829 

492 

28 

872 

88 

29 

.633 

1.942 

3.253 

4.665 

5.881 

7.200 

8.523 

9.852 

501 

29 

877 

89 

30 

.655 

1.964 

3.274 

4.587 

5.902 

7.222 

8.545 

9.874 

509 

30 

882 

90 

31 

.676 

1.986 

3.296 

4.609 

5.924 

7.243 

8.567 

9.896 

518 

31 

^7 

91 

32 

.698 

2.008 

3.318 

4.631 

5.947 

7.266 

8.589 

9.918 

526 

32 

892 

92 

33 

.720 

2.029 

3.340 

4.653 

5.968 

7.288 

8.611 

9.940 

534 

33 

897 

93 

34 

.742 

2.051 

3.362 

4.675 

5.990 

7.309 

8.633 

9.963 

542 

34 

901 

94 

35 

.763 

2.073 

3.384 

4.696 

6.013 

7.332 

8.656 

9.985 

550 

35 

906 

95 

36 

.785 

2.095 

3.406 

4.718 

6.034 

7.354 

8.677 

10.  007 

558 

36 

911 

96 

37 

.807 

2.116 

3.427 

4.741 

6.056 

7.375 

8.700 

10.  029 

S6G 

37 

916 

97 

38 

.829 

2.138 

3.449 

4.762 

6.078 

7.398 

8.722 

10.  051 

573 

38 

920 

98 

39 

.651 

2.161 

3.471 

4.784 

6.100 

7.420 

8.744 

10.  074 

581 

39 

925 

99 

40 

.873 

2.182 

3.495 

4.806 

6.122 

7.442 

8.766 

10.  096 

588 

40 

930 

100 

41 

.895 

2.204 

3.515 

4.828 

6.144 

7.464 

8.788 

10. 118 

595 

41 

934 

101 

42 

.916 

2.226 

3.537 

4.850 

6.166 

7.486 

8.810 

10. 141 

603 

42 

939 

102. 

43 

.938 

2.248 

3.559 

4.872 

6.188 

7.508 

8.833 

10. 162 

610 

43 

943 

103 

44 

.960 

2.269 

3.580 

4.894 

6.210 

7.530 

8.854 

10. 185 

617 

44 

948 

104 

45 

.982 

2.291 

3.602 

4.915 

6.232 

7.552 

8.877 

10.207 

624 

45 

953 

105 

46 

1.003 

2.313 

3.624 

4.938 

8.254 

7.574 

8.899 

10.  229 

631 

46 

957 

106 

47 

1.025 

2.335 

3.646 

4.960 

6.276 

7.596 

8.921 

10.  252 

637 

47 

962 

107 

48 

1.047 

2.357 

3.668 

4.981 

6.298 

7.618 

8.943 

10.  273 

644 

48 

966 

108 

49 

1.069 

2.379 

3.690 

5.003 

6.320 

7.640 

8.965 

10.  296 

651 

49 

971 

109 

50 

1.091 

2.401 

3.712 

5.025 

6.342 

7.662 

8.987 

10.318 

657 

50 

975 

110 

51 

1.113 

2.422 

3.733 

5.047 

6.364 

7.684 

9.010 

10.  340 

664 

51 

980 

111 

52 

1.135 

2.444 

3.755 

5.069 

6.385 

7.706 

9.031 

10.  363 

670 

52 

984 

112 

53 

1.156 

2.466 

3.776 

5.091 

6.408 

7.729 

9.054 

10.  384 

677 

53 

988 

113 

54 

1.178 

2.488 

3.799 

5.113 

6.430 

7.750 

9.076 

10.  407 

683 

54 

993 

114 

55 

1.200 

2.509 

3.821 

5.135 

6.451 

7.772 

9.098 

10.429 

690 

55 

997 

115 

56 

1.222 

2.532 

3.843 

5.157 

6. 474 

7.795 

9.120 

10.  451 

696 

56 

1001 

116 

57 

1.243 

2.554 

3.865 

5.179 

6.496 

7.816 

9.142 

10. 474 

702 

57 

1005 

117 

58 

1.265 

2.575 

3.886 

5.20U 

6.517 

7.839 

9.164 

10.  496 

708 

58 

1010 

118 

59 

1.287 

2.597 

3.900 

5.222 

6.540 

7.861 

9.187 

10.  518 

714 

59 

1014 

119 

60 

1.309 

2.619 

3.931 

5.245 

6.562 

7.882 

9.208 

10.540 

720 

60 

1018 

120 

ALTITUDE  TABLES. 


155 


Table  IX. — For  comjiuting  differences  of  altitude  from  angles  of  elevation  or  depression  (applicable  to 
scale  i;450(90)— Continued. 


h,  in  feet. 

D 

iH 

D 

Jh 

' 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

Scale 
divisions 

Feet. 

Scale 
divisions 

Feet. 

0 

10.  540 

11.  878 

13.  225 

14. 578 

15,942 

17,315 

18.  700 

20.  096 

00 

0 

720 

60 

1 

10.  563 

11.  901 

13.  247 

14.  601 

16,  964 

17, 338 

18,  723 

20. 119 

93 

1 

726 

61 

2 

10.  585 

11.  923 

13.  270 

14.  623 

15,  987 

17,  361 

18,  746 

20, 143 

131 

2 

732 

62 

S 

10.  607 

11.946 

13.  292 

14.  647 

16,  010 

17,  384 

18,  769 

20. 166 

101 

3 

73S 

63 

4 

10.  630 

11.  968 

13.315 

14.  669 

16.  033 

17,  407 

18,  792 

20. 190 

186 

4 

744 

64 

5 

lO.  651 

11.991 

13.  337 

14. 692 

16,  056 

17.  430 

18.813 

20.213 

208 

5 

750 

66 

6 

10.  674 

12.013 

13.  360 

14,  714 

16,  078 

17. 453 

18.  838 

20.236 

228 

6 

755 

66 

7 

10.  696 

12.  035 

13.  382 

14.  737 

16,  102 

17.476 

18,  862 

20,  260 

246 

7 

761 

67 

S 

10.718 

12.  069 

13.  405 

14.  760 

16,124 

17,499 

18,  885 

20,  283 

263 

8 

767 

68 

9 

10.  741 

12.  080 

13. 427 

14.782 

16. 147 

17.  522 

18,  908 

20.  307 

279 

9 

772 

69 

10 

10.763 

12. 103 

13.  450 

14.  806 

16, 170 

17.  545 

18,  931 

20,  330 

294 

10 

778 

70 

11 

10.  786 

12. 125 

13.  472 

14.  82S 

16, 192 

17,  568 

18,  955 

20,  353 

308 

11 

783 

71 

12 

10.  807 

12.147 

13.495 

14.  851 

16,  216 

17,  591 

18.  978 

20,  377 

322 

12 

789 

72 

18 

10. 830, 

12. 169 

13.  517 

14.  873 

16,  238 

17. 614 

19,  001 

20,  401 

335 

13 

794 

73 

14 

10.  862 

12. 192 

13. 540 

14.  896 

16,  261 

17.  637 

19.  024 

20,  424 

348 

14 

SOO 

74 

15 

10.  874 

12.214 

13.  562 

14.  918 

16,  284 

17,  660 

19,  048 

20,447 

S60 

15 

805 

75 

16 

10. 897 

12.  237 

13.  585 

14.  941 

16,  307 

17,  683 

19,  071 

20, 470 

372 

16 

811 

76 

17 

10.  919 

12.  259 

13. 607 

14.  964 

16,  330 

17,  706 

19,  094 

20.494 

383 

17 

816 

77 

18 

10.  941 

12.  282 

13.  630 

14.  986 

16,  353 

17, 729 

19, 117 

20,  518 

394 

18 

821 

78 

19 

10.  963 

12.  304 

13.  662 

15. 009 

16,  375 

17,  752 

19, 142 

20.  541 

405 

19 

826 

79 

20 

10.  986 

12.  326 

13.  676 

16.  031 

16,  398 

17,  775 

19. 164 

20.  564 

416 

20 

832 

80 

21 

11.008 

12.  349 

13.  697 

15. 055 

16.  421 

17,  798 

19. 187 

20,  588 

426 

21 

837 

81 

22 

11.030 

12.  371 

13.  720 

15.  077 

16,444 

17,  821 

19,  210 

20,  611 

436 

22 

842 

82 

23 

11.  053 

12.  394 

13.  742 

15.100 

16, 467 

17,  845 

19,  234 

20, 635 

446 

23 

847 

S3 

21 

11.075 

12.416 

13.  766 

15. 123 

16.  489 

17,  867 

19,  257 

20.  659 

455 

24 

852 

84 

25 

11.  097 

12.  439 

13.  787 

15,  145 

16.513 

17,  890 

19.  280 

20,  682 

465 

25 

857 

85 

26 

11.119 

12.461 

13.810 

15,168 

16,535 

17,  914 

19,  303 

20,  705 

474 

26 

86S 

80 

27 

11. 142 

12.  484 

13.  833 

15. 190 

16,  558 

17,  937 

19.  327 

20,  723 

483 

27 

867 

87 

2S 

11. 164 

12.505 

13.865 

15.  214 

16, 581 

17,  959 

19.350 

20.  752 

492 

28 

872 

88 

29 

U.  186 

12.  528 

13.  878 

15. 236 

16,  604 

17.  983 

19,373 

20,  776 

501 

29 

877 

89 

30 

11.  209 

12.  550 

13.  009 

15.  259 

16,  627 

18,  006 

19,  396 

20,  799 

509 

30 

882 

90 

SI 

11.  231 

12.  .573 

13.  923 

15.282 

16,  650 

18,  029 

19.  420 

20,  823 

518 

31 

887 

91 

32 

11.  254 

12.  .595 

13. 945 

15,  304 

16,  673 

18,  052 

19,443 

20, 846 

526 

32 

892 

92 

S3 

11.  275 

12.  618 

13.  968 

15,  327 

16,  696 

18,  075 

19,  466 

20,  869 

534 

32 

897 

93 

34 

11.  298 

12.  640 

13.  990 

15,  349 

16,  719 

18. 097 

19. 489 

20,  893 

542 

34 

901 

94 

35 

11.  320 

12.j60i 

14.  013 

16,  373 

16,  741 

18, 121 

19,  513 

20,  917 

550 

35 

906 

95 

S6 

11.  343 

12.  683 

14.  o;i6 

15,395 

16,  765 

18, 145 

19,536 

20, 940  . 

55S 

36 

911 

96 

37 

11. 366 

12.  708 

14.  059 

15,  418 

16.  787 

18,167 

19,  559 

20. 964 

566 

37 

916 

97 

ss 

11.387 

12.  730 

14.  081 

15,441 

16.810 

18. 190 

19,  582 

20,  987 

573 

38 

920 

98 

39 

11. 410 

12.  753 

14. 104 

15.  463 

16.  833 

18,  214 

19,  606 

21.  Oil 

581 

39 

925 

99 

40 

11.432 

12.775 

14.  126 

15,  486 

16.856 

18,  237 

19.  629 

21,034 

588 

40 

930 

loo 

41 

11.454 

12.  707 

u.  m 

15,509 

16,  870 

IS,  260 

19,  652 

21.  0.58 

595 

41 

934 

101 

42 

11.476 

12. 8J0 

14.  1 71 

15  532 

16,  902 

18  283 

19,  (176 

21,082 

603 

42 

989 

102 

43 

11.499 

12.  842 

14. 194 

15.  ,554 

16,  925 

18,306 

19.  699 

31,  105 

610 

43 

943 

103 

44 

11.521 

12.865 

14.  216 

15,577 

16,  948 

18,329 

19,723 

21, 120 

017 

44 

948 

104 

45 

11.  543 

12.  887 

14.  239 

16,  600 

16,  971 

18,  352 

19,  746 

21, 152 

624 

45 

953 

105 

46 

11.  566 

12,910 

14.  262 

15.  622 

16.  993 

18.  376 

19,  769 

21. 175 

631 

46 

957 

106 

-  47 

11. 588 

12.932 

14.  284 

15,  646 

17, 017 

18,  399 

19,  792 

21, 199 

637 

47 

962 

107 

4>l 

11.611 

12.  955 

14.307 

16,  668 

17,039 

18,  421 

19,816 

21, 223 

644 

48 

966 

108 

49 

11.633 

12.  977 

14.  329 

15,  691 

17.  062 

18,  445 

19, 839 

21,  247 

651 

49 

971 

109 

50 

11.  6.55 

13.  000 

14.  352 

15,  714 

17.  086 

18,  468 

19,  862 

21.  270 

657 

50 

975 

110 

51 

11.  677 

13.  022 

14.  374 

15,  736 

17, 108 

18.491 

19. 886 

21.  293 

664 

51 

980 

lU 

52 

11.700 

13.  045 

14.  398 

15,760 

17,131 

18,  514 

19,  909 

21.  317 

670 

52 

984 

112 

53 

11.722 

13.  067 

14.  420 

15,  782 

17. 154 

18,  538 

19,  933 

21.  340 

677 

53 

988 

113 

54 

11.  745 

13.  090 

14.  443 

16.805 

17. 177 

18,  560 

19.  956 

21.  364 

683 

54 

993 

114 

55 

11.767 

13.112 

14.  465 

15,  828 

17,  200 

18,  583 

19,  979 

21,388 

690 

55 

997 

115 

56 

11.789 

13. 135 

14.  488 

15,  850 

17,  223 

18,  607 

20.  002 

21,412 

696 

56 

1001 

116 

57 

lx.812 

13. 157 

14.  510 

15,  873 

17,  246 

18,  630 

20,  026 

21,  435 

702 

57 

1005 

117 

58 

11.834 

13. 180 

14.  533 

15,  896 

17,  269 

18,663 

20,  050 

21.  459 

708 

69 

1010 

118 

59 

11.  857 

13.  202 

14.  556 

15,  919 

17.  292 

18,  676 

20.  073 

21,482 

714 

58 

1014 

119 

60 

11.  878 

13.  225 

14.578 

16,  942 

17. 315 

IS.  700 

20.096 

21.  506 

720 

60 

1018 

120 

156 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


-For  computing  differences  of  altitude  from  angles  of  elevation  or  depreasion  (applicahle  to  scale 
of  1:30000). 
[Prepared  Ipy  K.  S.  "Woodward. 1 


■r^-^v  e  ixjx    i„      C +D7i,4-/ia  for  ans'les  of  elevation. 

Ditference  of  altitnde=  J  Iu,,;_|.,,.^  for  „„|ies  of  dopressioi 


opressiou. 
^distance  in  scale  divisions  E>j'ineli  each;  (t  =  angle  of  elevation  or  depression ;  7i,=50feetx  tan  a;  7i2  =  correction 
for  curvature  and  refraction. 

Arguuient  for  Ai  is  a,-  argument  for  /(•.»  is  D. 


A,  in  feet. 

D 

ft2 

D 

7i2 

' 

0=> 

1° 

20 

3° 

4° 

5° 

«o 

JO 

Scale 
divisions. 

Feet. 

Scale 
divisions. 

Feet. 

0 

.000 

.873 

1.746 

2.620 

3.496 

4  374 

5.255 

6139 

000 

0 

1080 

60 

1 

.014 

.887 

1.760 

2.635 

3.  511 

4  389 

5.270 

6.154 

130 

1 

1089 

61 

2 

.029 

.902 

1.775 

2.649 

3.525 

4  403 

5.284 

6.109 

197 

2 

1098 

62 

3 

.043 

.916 

1.789 

2.664 

3.540 

4  418 

5.299 

6. 183 

243 

3 

1107 

63 

4 

.058 

.931 

1.804 

2.678 

3.555 

4  433 

5.314 

6.198 

270 

4 

1116 

64 

5 

.072 

.945 

1.819 

2.693 

3.569 

4.447 

5.328 

6.213 

312 

5 

1124 

65 

6 

.087 

.960 

1.833 

2.708 

3.584 

4  462 

5.343 

6.  228 

342 

6 

1133 

66 

7 

.102 

.974 

1.848 

2.722 

.  3.  598 

4  477 

5.358 

6.242 

309 

7 

1141 

67 

g 

.116 

.989 

1.862 

2.737 

3.613 

4  491 

5.373 

6.257 

394 

8 

1150 

68 

9 

.131 

1.003 

1.877 

2.751 

3.G28 

4  506 

5.387 

6.272 

418 

9 

1158 

69 

10 

.145 

1.018 

1.891 

2.706 

3.642 

4  521 

5.402 

6.287 

441 

10 

1107 

70 

11 

.160 

1.033 

1.906 

2.781 

3.057 

4  535 

5.417 

6.301 

463 

11 

1175 

71 

12 

.174 

1.047 

1.921 

2.795 

3.072 

4.550 

5.431 

6.316 

483 

12 

11, S3 

72 

13 

.189 

1.0B2 

1.935 

2.810 

3.686 

4.565 

5.446 

6.331 

503 

13 

1191 

73 

14 

•    .203 

1.076 

1.950 

2.824 

3.701 

4  579 

5.461 

6.346 

522 

14 

1199 

74 

13 

.218 

1.091 

1.964 

2.839 

3.715 

4.594 

5.476 

6  361 

540 

15 

1208 

75 

16 

.232 

1.105 

1.979 

2.854 

3.730 

4. 609 

5.490 

6.375 

558 

16 

1216 

76 

17 

.247 

1.120 

1.993 

2.868 

3.745 

4  623 

5  505 

6.390 

575 

17 

1234 

77 

IS 

.262 

1.134 

2.008 

2.883 

3.759 

4.  638 

5.  £20 

6  405 

592 

18 

1231 

78 

19 

.276 

1.149 

2.023 

2.897 

3.774 

4  653 

5.535 

6.420 

608 

19 

1339 

79 

20 

.291 

1.164 

2.037 

2.912 

3.789 

4.667 

5.549 

6  434 

624 

20 

1247 

80 

21 

.305 

1.178 

2.052 

2.927 

3.  803 

4  682 

5.564 

6  449 

639 

21 

1255 

81 

22 

.320 

1.193 

2.066 

2.941 

3.818 

4.097 

5.579 

6.464 

654 

22 

1363 

83 

23 

.334 

1.207 

2.081 

2.956 

3.832 

4.711 

5.593 

6.479 

669 

23 

1270 

83 

24 

.349 

1.222 

2.095 

2.970 

3.847 

4  726 

5.608 

6.494 

683 

24 

1378 

84 

25 

.363 

1.236 

2.110 

2.985 

3.862 

4  741 

5.623 

6.508 

697 

25 

1286 

85 

26 

.378 

1.250 

2.125 

2.999 

3.876 

4  755 

5.638 

6  523 

711 

26 

1293 

86 

27 

.392 

1.265 

2.139 

3.014 

3.891 

4  770 

5.652 

6.538 

725 

27 

1301 

87 

28 

.407 

1.280 

2.154 

3.029 

3.906 

4  785 

5.667 

6.  553 

738 

28 

1308 

88 

29 

.422 

1.294 

2.168 

3.043 

3.920 

4  800 

5.682 

6.568 

751 

29 

1315 

89 

30 

.436 

1.309 

2.183 

3.058 

3.935 

4  814 

5.697 

6  582 

764 

30 

1333 

90 

31 

.451 

1.324 

2.197 

3.072 

3.949 

4  829 

5.711 

6.597 

776 

31 

1330 

91 

33 

.465 

1.338 

2.212 

3.087 

3.  964 

4  844 

5.726 

6.612 

789 

32 

1337 

92 

33 

.480 

1.353 

2.227 

3.102 

3.979 

4  858 

5.741 

6.627 

801 

33 

1345 

93 

34 

.494 

1.367 

2.241 

3.116 

3.993 

4  873 

5.755 

6.642 

813 

34 

1353 

94 

35 

.509 

1.382 

2.256 

3.131 

4.008 

4  888 

5.770 

6.656 

825 

35 

1359 

95 

36 

.523 

1.396 

2.270 

3. 145 

4.  023 

4  902 

5.785 

6.671 

837 

36 

1366 

96 

37 

.538 

1.411 

2.285 

3.160 

4  037 

4.917 

5.800 

6.686 

848 

37 

1373 

97 

3S 

.552 

1.425 

2.289 

3.175 

4  052 

4.932 

5.814 

6.701 

860 

38 

1380 

98 

39 

.567 

1.440 

2.314 

3.189 

4.067 

4  946 

5.829 

6  716 

871 

39 

1387 

99 

40 

.582 

1.455 

2.329 

3.204 

4.081 

4.961 

5.844 

6. 730 

882 

40 

1394 

100 

41 

.596 

1.469 

2.343 

3.218 

4.096 

4  976 

5.859 

6.745 

893 

41 

1401 

101 

42 

.611 

1.4C4 

2.358 

3.233 

4110 

4.990 

5.873 

6.760 

904 

42 

1408 

102- 

43 

.625 

1.498 

2.372 

3.248 

4.125 

5.005 

5.888 

6.775 

914 

43 

1415 

103 

44 

.640 

1.513 

2.387 

3.262 

4.  140 

5.020 

5.  903 

6.790 

925 

44 

1422 

104 

45 

.654 

1.527 

2.401 

3.277 

4154 

5.034 

5.918 

6.804 

935 

45 

1429 

105 

46 

.669 

1.542 

2.416 

3.292 

4.169 

5.049 

5.932 

6  819 

946 

46 

1436 

106 

47 

.683 

1.657 

2.431 

3.306 

4184 

5.064 

5.947 

6  834 

95G 

47 

1442 

107 

48 

.698 

1.571 

2.445 

3.321 

4198 

5.079 

5.962 

6  849 

966 

48 

1449 

108 

49 

.712 

1.586 

2.460 

3.335 

4  213 

5.093 

5.977 

6.864 

976 

49 

1456 

109 

60 

.727 

1.600 

2.474 

3.350 

4.228 

5.108 

5.991 

6.879 

980 

50 

1462 

110 

51 

.742 

1.615 

2.489 

3.365 

4.  242 

5.123 

6.006 

6.893 

996 

5] 

1469 

111 

52 

.756 

1.629 

2.503 

3.379 

4  257 

5.137 

6.021 

6  908 

1006 

52 

1476 

112 

53 

.771 

1.644 

2.517 

3.394 

4  272 

5.152 

6.036 

6  923 

1015 

53 

1482 

113 

54 

.785 

1.658 

2.533 

3.408 

4  286 

5.167 

6.050 

6  938 

1025 

54 

1489 

114 

55 

.800 

1.  673 

2.547 

3.423 

4  301 

5.181 

6.065 

6.953 

1034 

55 

1495 

115 

56 

.814 

1.688 

2.562 

3.438 

4.316 

5.196 

6.080 

6  967 

1043 

56 

1502 

116 

57 

.829 

1.702 

2.576 

3.452 

4  330 

5.211 

6.095 

6.982 

1053 

57 

1508 

117 

58 

.843 

1.717 

2.591 

3.467 

4  345 

5.226 

6.109 

6.907 

1062 

58 

1515 

118 

59 

.858 

1.731 

2.606 

^  3.  481 

4.360 

5.240 

6.124 

7.012 

1071 

59 

1521 

119 

60 

.873 

1.746 

2.620 

3.496 

4  374 

5.255 

6.139 

7.027 

1080 

60 

1527 

120 

ALTITUDE  TABLES. 


157 


-For  com]}utiiig  differences  of  altitude  from  aiu/les  of  elevation  or  depression  (applicable  to  scale 
of  1:  30000— Contiuued. 


ft,  in  feet. 

D 

1h 

D 

h^ 

' 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

Scale 
[livisions. 

Feet. 

Scale 
divisions. 

Feet. 

0 

7.027 

7.919 

8.816 

9.719 

10.  628 

11.543 

12.  466 

13.  397 

000 

0 

1080 

60 

1 

7.042 

7.934 

8.831 

9.734 

10. 643 

11.  558 

12.482 

13.  413 

139 

1 

1089 

61 

2 

7.056 

7.949 

8.846 

9.749 

10.  658 

11. 574 

12.497 

13.428 

197 

2 

1098 

62 

3 

7.071 

7.964 

8.861 

9.704" 

10.  673 

11.  589 

12.  51? 

13.444 

242 

3 

1107 

63 

4 

7.086 

7.979 

8.876 

9.779 

10.  688 

11. 604 

12. 528 

13. 460 

279 

4 

1116 

64 

o 

7.101 

7.994 

8.891 

9.794 

10.  704 

11.  620 

12.  543 

13. 475 

312 

g 

1124 

65 

6 

7.116 

8.008 

8.906 

9.809 

10.  719 

11.635 

12,  559 

13.  491 

342 

6 

1133 

66 

7.131 

8.023 

8.921 

9.824 

.  10.  734 

11.  650 

12.  574 

13.506 

369 

7 

1141 

67 

8 

7.145 

8.038 

8.936 

9.840 

10.  749 

11.  666 

12. 590 

13.  522 

394 

8 

1150 

68 

9 

7.160 

8.053 

8.951 

9,855 

10.  764 

11.  681 

12.  605 

13.  538 

418 

9 

1158 

69 

10 

7.175 

8.068 

8.966 

9.  870 

10.7.SO 

11.696 

12. 621 

13.  553 

441 

10 

1167 

70 

11 

7.190 

8.083 

8.981 

9.885 

10.795 

11.712 

12.  636 

13.  569 

462 

11 

1175 

71 

12 

7.205 

8.098 

8.996 

9.900 

10.  810 

11. 727 

12,  652 

13. 584 

483 

12 

1183 

72 

13 

7.220 

8.113 

9.011 

9.915 

10.  825 

11.  742 

12,  667 

13.  600 

503 

13 

1191 

73 

14 

7.235 

8.128 

9.026 

9.  930 

10.841 

11.  758 

12,683 

13.  616 

522 

14 

1199 

74 

15 

7.249 

8.143 

9.041 

9.945 

10.856 

11.  773 

12,  698 

13.  631 

540 

15 

1208 

75 

16 

7.264 

8.158 

9.056 

9.960 

10.871 

11.789 

12.  714 

13.  647 

558 

16 

1216 

76 

17 

7.279 

8. 173 

9.071 

9.976 

10.886 

11.804 

12.  729 

13. 663 

575 

17 

1224 

77 

18 

7.294 

8.188 

9.086 

9.991 

10.  902 

11.  819 

12.745 

13.  678 

592 

18 

1231 

78 

19 

7.309 

8.202 

9.101 

10.006 

10.  917 

11.  835 

12. 761 

13.  694 

608 

19 

1239 

79 

20 

7.324 

8.217 

9.116 

10.021 

10.  932 

11,  850 

12.  776 

13.709 

624 

20 

1247 

80 

21 

7.339 

8.232 

9.131 

10.  036 

10.  947 

11,  865 

12.  791 

13. 725 

639 

21 

1255 

81 

22 

7.353 

8.247 

9.146 

10.  051 

10.  962 

11.881 

12.  807 

13. 741 

654 

22 

1263 

82 

23 

7.368 

8.202 

9.161 

10. 066 

10.978 

11.896 

12.  822 

13.756 

669 

23 

1270 

83 

21 

7.383 

8.277 

9.176 

10.  082 

10. 993 

11.911 

12. 838 

13.772 

683 

24 

1278 

84 

25 

7.398 

8.292 

9.191 

10.097 

11. 008 

11.927 

12.  853 

13.788 

697 

25 

1286 

85 

26 

7.413 

8.307 

9.207 

10. 112 

11.  023 

11.  942 

12. 869 

13,  803 

711 

26 

1293 

86 

27 

7.428 

8.322 

9.222 

10. 127 

11,  039 

11,958 

12. 884 

13. 819 

725 

27 

1301 

87 

28 

7.443 

8.337 

9.237 

10.142 

11.  054 

11.  973 

12. 900 

13. 835 

738 

28 

1308 

88 

29 

7.457 

8.352 

9.252 

10, 157 

11.  069 

11. 988 

12.915 

13,  860 

751 

29 

1315 

89 

30 

7.472 

8.367 

9.267 

10. 172 

11.  084 

12.  004 

12.  931 

13,866 

764 

30 

1323 

90 

31 

7.487 

8.382 

9. 282 

10. 18B 

11. 100 

12,019 

12,  946 

13.  882 

776 

31 

1330 

91 

32 

7.502 

8.397 

9.297 

10.203 

11.115 

12.  034 

12,  962 

13,  897 

789 

32 

1337 

92 

33 

7.517 

8.412 

9.312 

10.218 

11. 130 

12,030 

12,  077 

13,913 

SOI 

33 

1345 

93 

3* 

7.532 

8.427 

9.327 

10.233 

11. 146 

12,  065 

12,993 

13.929 

813 

34 

1352 

94 

35 

7.547 

8.442 

9.342 

10.248 

11. 161 

12,081 

13.  008 

13.944 

825 

35 

13.59 

95 

36 

7.562 

8.457 

9.3.57 

10.263 

11. 176 

12.  096 

13, 024 

13.  960 

837 

36 

1366 

96 

37 

7.576 

8.472 

9.372 

10.278 

11. 191 

12,  111 

13.  039 

13. 976 

848 

37 

1373 

97 

38 

7.591 

8.487 

9.387 

10.  294 

11.207 

12. 127 

13.  055 

13. 991 

860 

38 

1380 

98 

39 

7.606 

8.502 

9.402 

10.  309 

11.222 

12. 142 

13,070 

14.007 

871 

39 

■     1387 

99 

40 

7.621 

8.516 

9.417 

10.  324 

11.  237 

12.  158 

13.  086 

14.  023 

882 

40 

1394 

100 

41 

7.636 

8.531 

9.432 

10.  339 

11.  252 

12. 173 

13. 101 

14.038 

893 

41 

1401 

101 

42 

7.651 

8.546 

9.447 

10.354 

11,  268 

12. 188 

13.117 

]4.0a4 

904 

42 

1408 

102 

43 

7.666 

8.561 

9.462 

10.369 

11.  283 

12. 204 

13. 133 

14. 070 

914 

43 

1415 

103 

44 

7.681 

8.576 

9.477 

10.  385 

11,  298 

12.  219 

13. 148 

14.  086 

925 

44 

1422 

104 

45 

7.695 

8.591 

9.493 

10.400 

11.314 

12.  235 

13. 164 

14.101 

935 

45 

1429 

105 

.46 

7.710 

8.606 

9.508 

10. 415 

11.  329 

12.250 

13. 179 

14.117 

946 

46 

1436 

106 

47 

7.725 

8.621 

9.523 

10.431 

U.344 

12.  266 

13. 195 

14. 133 

936 

47 

1442 

107 

48 

7.740 

8.636 

9.538 

10.445 

11.  359 

12.  281 

13.210 

14148 

966 

48 

1449 

108 

49 

7.755 

8.651 

9.553 

10. 460 

11.375 

12.  296 

13.226 

14. 164 

976 

49 

1456 

109 

50 

7.770 

8.666 

9.568 

10. 476 

11. 390 

12.312 

13.  241 

14.180 

986 

50 

1462 

110 

51 

7.785 

8.681 

9.583 

10. 491 

11.405 

12.327 

13.  257 

14. 195 

996 

51 

1469 

111 

62 

7.800 

8.696 

9.598 

10.506 

11.421 

12,  343 

13.  273 

14.211 

1006 

52 

1476 

112 

63 

7.815 

■   8.711 

9.613 

10.521 

11.  436 

12,  358 

13,  288 

14.  227 

1015 

53 

1482 

113 

54 

7.830 

8.72D 

9.628 

10. 536 

11.  451 

12.373 

13,  304 

14.243 

1025 

54 

1489 

114 

56 

7.844 

8.741 

9.643 

10. 552 

11. 467 

12.  389 

13.  319 

14.258 

1034 

55 

1495 

115 

56 

7.859 

8.756 

9.658 

10.  567 

11.482 

12.  404 

13.  335 

14.  274 

1043 

56 

1502 

116 

57 

7.874 

8.771 

9.673 

10.  5S2 

11. 497 

12.  420 

13.  350 

14.  290 

1053 

57 

1508 

117 

68 

7.  889 

8.786 

9.689 

10.  597 

11.513 

12.435 

13.366 

14.  306 

1062 

58 

1515 

118 

59 

7.904 

8.801 

9.704 

10.  612 

11.  528 

12.451 

13. 382 

14.321 

1071 

59 

1521 

119 

60 

7.919 

8.816 

9.719 

10.028 

11.  543 

12.406 

13.  397 

14.337 

1080 

60 

1527 

120 

158 


A  MANUAL  OF  TOPOGEAPHIO  METHODS. 


Table  XI. — Differences  of  altUude 


[Prepared  by 
Computed  from  the  formula  A  ^  D  sin  a  cos  a,  in  which  D  is  the  observed  distance  of  the 


D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

" 

5G0 

5S0 

600 

620 

640 

660 

6S0 

700 

720 

740 

760 

780 

800 

820 

0    01 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0   02 

0.3 

0.3 

0.3 

0.4 

0.4. 

0.4 

0.4 

0.4 

.     0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0    03 

0.5 

0.5 

0.5 

0.5 

0.6 

0.6 

O.B 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.7 

U   01 

0.6 

0.7 

0.7 

0.7 

0.7 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0.9 

1.0 

0  05 

0.8 

0.8 

0.9 

0.9 

0.9 

1.0 

1.0 

1.0 

1.0 

1.1 

1.1 

1.1 

1.2 

1.2 

0  06 

1.0 

1.0 

1.1 

1.1 

1.1 

1.2 

1.2 

1.2 

1.3 

1.3 

1.3 

1.4 

1.4 

1.4 

0   07 

1.1 

1.2 

1.2 

1.3 

1.3 

1.3 

1.4 

1.4 

1.5 

1.5 

1.6 

1.6 

1.6 

1.7 

0   08 

1.3 

1.4 

1.4 

1.4 

1.5 

1.5 

1.6 

1.6 

1.7 

1.7 

1.8 

1.8 

1.9 

1.9 

0   09 

1.5 

1.5 

1.6 

1.6 

1.7 

1.7 

1.8 

1.8 

1.9 

1.9 

2.0 

2.0 

2.1 

2.1 

0  10 

1.6 

1.7 

1.7 

1.8 

1.9 

1.9 

2.0 

2.0 

2.1 

2.2 

2.2 

2.3 

2.3 

2.4 

0    11 

1.8 

1.9 

1.9 

2.0 

2.0 

2.1 

2.2 

2.2 

2.3 

2.4 

2.4 

2.5 

2.6 

2.6 

0  12 

2.0 

2.0 

2.1 

2.2 

2.2 

2.3 

2.4 

2.4 

2.5 

2.6 

2.7 

2.7 

2.8 

2.9 

0  13 

2.1 

2.2 

2.3 

2.3 

2.4 

2.5 

2.6 

2.6 

2.7 

2.8 

2.9 

2.9 

3.0 

3.1 

0  14 

2.3 

2.4 

2.4 

2.5 

2.6 

2.7 

2.8 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.3 

0  IS 

2.4 

2.5 

2.6 

2.7 

2.8 

2.9 

3.0 

3.1 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

0  16 

2.6 

2.7 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.3 

3.4 

3.5 

3.6 

3.7 

3.8 

0  17 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

3.8 

3.9 

4.0 

4.1 

0   18 

2.9 

3.0 

3.1 

3.2 

3.4 

3.5 

3.6 

3.7 

3.8 

3.9 

4.0 

4.1 

4.2 

4.3 

0  19 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.8 

3.9 

4.0 

4.1 

4.2 

4.3 

4.4 

4.5 

0  20 

3.3 

3.4 

3.5 

3.6 

3.7 

3.8 

4.0 

4.1 

4.2 

4.3 

4.4 

4.5 

4.7 

4.8 

0  21 

3.4 

3.5 

3.7 

3.8 

3.9 

4.0 

4.2 

4.3 

4.4 

4.5 

4.6 

4.8 

4.9 

5.0 

0   22 

3.6 

3.7 

3.8 

4.0 

4.1 

4.2 

4.4 

4.5 

4.6 

4.7 

4.9 

5.0 

5.1 

5.2 

0   23 

3.7 

3.9 

4.0 

4.1 

4.3 

4.4 

4.5 

4.7 

4.8 

5.0 

5.1 

5.2 

5.4 

5.5 

0  24 

3.9 

4.0 

4.2 

4.3 

4.5 

4.0 

4.7 

4.9 

5.0 

5.2 

5.3 

5.4 

5.6 

5.7 

0  25 

4.1 

4.2 

4.4 

4.5 

4.7 

4.8 

4.9 

5.1 

5.2 

5.4 

5.5 

5.7 

5.8 

6.0 

0  26 

4.2 

4.4 

4.5 

4.7 

4.8 

5.0 

5.1 

5.3 

5.4 

5.6 

5.7 

5.9 

6.0 

6.2 

U  27 

4.4 

4.6 

4.7 

49 

5.0 

5.2 

5.3 

5.5 

5.7 

5.8 

6.0 

6.1 

6.3 

6.4 

0  28 

4.6 

4.7 

4.9 

5.0 

5.2 

5.4 

5.5 

5.7 

5.9 

6.0 

6.2 

6.3 

6.5 

6.7 

0  29 

4.7 

4.9 

5.1 

5.2 

5.4 

5.6 

5.7 

5.9 

6.1 

6.2 

6.4 

6.6 

6.8 

6.9 

0  30 

4.9 

5.1 

5.2 

5.4 

5.6 

5.8 

5.9 

6.1 

6.3 

6.5 

6.6 

6.8 

7.0 

7.2 

0  35 

5.7 

5.9 

6.1 

6.3 

6.5 

6.7 

6.9 

7.1 

7.3 

7.5 

7.7 

7.9 

8.1 

8.4 

0   40 

6.5 

6.7 

7.0 

7.2 

7.4 

7.7 

7.9 

8.1 

8.4 

8.6 

8.8 

9.1 

9.3 

9.5 

0   45 

7.3 

7.6 

7.9 

8.1 

8.4 

8.6 

8.9 

9.2 

9.4 

9.7 

9.9 

10.2 

10.5 

10.7 

0   50 

8.1 

8.4 

8.7 

9.0 

9.3 

9.6 

9.9 

10.2 

10.5 

10.8 

11.1 

11.3 

11.6 

11.9 

0   55 

9.0 

9.3 

9.6 

9.9 

10.2 

10.6 

10.9 

11.2 

11.5 

11.8 

12.2 

12.5 

12.8 

13.1 

1  00 

9.8 

10.1 

111.5 

10.8 

11.2 

11.5 

11.9 

12.2 

12.6 

12.9 

13.3 

13.6 

14.0 

14.3 

1    10 

11.4 

11.8 

12.2 

12.6 

13.0 

13.4 

13.8 

14.3 

14.7 

15.1 

15.5 

15.9 

16.3 

16.7 

1  20 

13.0 

13.5 

14.0 

14.4 

14.9 

15.4 

15.8 

16.3 

16.7 

17.2 

17.7 

18.1 

18.6 

19.1 

1  30 

14.7 

15.2 

15.7 

16.2 

16.7 

17.3 

17.8 

18.3 

18.8 

19.4 

19.9 

20.4 

20.9 

21.5 

1   40 

10.3 

16.9 

17.4 

18.0 

18.6 

19.2 

19.8 

20.3 

20.9 

21.5 

22.1 

22.7 

23.3 

23.8 

1   50 

17.9 

18.5 

19.2 

19.8 

20.5 

21.1 

21.7 

22.4 

23.0 

23.7 

24.3 

24.9 

25.6 

26.2 

2  00 

19.5 

20.2 

20.9 

21.6 

22.3 

23.0 

23.7 

24.4 

25.1 

25.8 

26.5 

27.2 

27.9 

28.6 

2   10 

21.2 

21.9 

22.7 

23.4 

24.2 

24.9 

25.7 

26.4 

27.2 

28.0 

28.7 

29.5 

30.2 

31.0 

2   20 

22.8 

23.6 

24.4 

25.2 

26.0 

26.8 

27.7 

28.5 

29.3 

30.1 

30.9 

31.7 

32.5 

33.4 

2   30 

24.4 

25.3 

26.1 

27.0 

27.9 

28.8 

29.6 

30.5 

31.4 

32.2 

33.1 

34.0 

34.9 

35.7 

2   40 

26.0 

27.0 

27.9 

28.8 

29.7 

30.7 

31.6 

32.5 

33.5 

34.4 

35.3 

36.3 

37.2 

38.1 

2  50 

27.6 

28.6 

29.6 

30.6 

31.6 

32.0 

33.6 

34.6 

35.5 

36.5 

37.5 

38.5 

39.5 

40.5 

'8  00 

29.3 

30.3 

31.4 

32.4 

33.4 

34.5 

35.5 

36.6 

37.6 

38.7 

39.7 

40.8 

41.8 

42.9 

1  00 

39.0 

40.4 

41.8 

43.1 

44.6 

45.9 

47.3 

48.7 

50.1 

51.5 

52.9 

54.3 

55.7 

57.1 

5  00 

48.6 

50.4 

52.1 

53.8 

55.6 

57.3 

59.0 

60.8 

62.5 

64.2 

66.0 

67.7 

69.5 

71.2 

D 

D 

D 

D 

D 

D 

D 

D 

» 

D 

D 

D 

D 

D 

560 

580 

000 

620 

640 

660 

6S0 

700 

720 

740 

760 

780 

800 

820 

ALTITUDE  TABLES. 


159 


from  telemeter  measures. 


R.S.  Woodward.] 
telemeter  staff,  a  is  the  £ 


gle  of  elevation  or  depression,  and  h  is  the  difference  in  height. 


D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

» 

D 

D 

It 

840 

860 

880 

900 

920 

940 

960 

980 

1,000 

1,100 

1,200 

1,S00 

1,400 

1,500 

2,000 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.6 

0.5 

0.5 

0.5 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.6 

0.7 

0.8 

0.8 

0.9 

1.2 

0.7 

0.7 

0.8 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.7 

1.0 

1.0 

1.0 

1.0 

1.1 

1.1 

1.1 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

2.3 

1.2 

1.2 

1.3 

1.3 

1.3 

1.4 

1.4 

1.4 

1.5 

1.6 

1.7 

1.9 

2.0 

2.2 

2.9 

1.5 

1.5 

1.5 

1.6 

1.6 

1.6 

1.7 

1.7 

1.7 

1.9 

2.1 

2.3 

2.4 

2.6 

3.5 

1.7 

1.8 

1.8 

1.8 

1.9 

3.9 

2.0 

2.0 

2.0 

2.2 

2.4 

2.7 

2.9 

3.1 

4.1 

2.0 

2.0 

2.1 

2.1 

2.1 

2.2 

2.2 

2.3 

2.3 

2.6 

2.8 

3.0 

3.3 

3.5 

4.7 

2.2 

2.3 

2.3 

2.4 

2.4 

2.5 

2.5 

2.6 

2.6 

2.9 

3.1 

3.4 

3.7 

3.9 

5.2 

2.4 

2.5 

2.6 

2.6 

2.7 

2.7 

2.8 

2.9 

2.9 

3.2 

3.5 

3.8 

4.1 

4.4 

5.8 

2.7 

2.8 

2.8 

2.9 

2.9 

3.0 

3.1 

3.1 

3.2 

3.5 

3.8 

4  2 

4  5 

4  8 

6.4 

2.9 

3.0 

3.1 

3.1 

3.2 

3.3 

3.4 

3.4 

3.5 

3.8 

4  2 

4  5 

4  9 

5.2 

7.0 

3.2 

3.3 

3.3 

3.4 

3.5 

3.6 

3.6 

3.7 

3.8 

4  2 

4  5 

4  9 

5.3 

5.7 

7.6 

3.4 

8.5 

3.6 

3.7 

3.7 

3.8 

3.9 

4.0 

4.1 

4.5 

4.9 

5.3 

5.7 

6.1 

8.1 

3.7 

3.7 

3.8 

3.9 

4.0 

41 

4.2 

4.3 

4  4 

4  8 

5.2 

5.7 

6.1 

6.5 

8.7 

3.9 

4  0 

4.1 

4.2 

4.3 

4  4 

4  5 

4.6 

4  7 

5.1 

5.6 

6.0 

6.5 

7.0 

9.3 

4.2 

4  3 

4.4 

4  5 

4.6 

4  7 

4  8 

4  9 

5.0 

5.4 

5.9 

6.4 

6.9 

7.4 

9.9 

4.4 

4.5 

4.6 

4  7 

4  8 

4  9 

5.0 

5.1 

5.2 

5.8 

6.3 

6.8 

7.3 

7.9 

10.5 

4  6 

4  8 

4.9 

5.0 

5.1 

5.2 

5.3 

5.4 

5.5 

6.1 

6.6 

7.2 

7.7 

8.3 

11.1 

4.9 

5.0 

5.1 

5.2 

5.4 

5.5 

5.6 

5.7 

5.8 

6.4 

7.0 

7.5 

8.1 

8.7 

11.6 

5.1 

5.3 

5.4 

5.5 

5.6 

5.7 

5.9 

6.0 

6.1 

6.7 

7.3 

7.9 

8.6 

9.2 

12.2 

5.4 

5.5 

5.6 

5.8 

5.9 

6.0 

6.1 

6.3 

6.4 

7.0 

7.7 

8.3 

9.0 

9.6 

12.8 

5.6 

5.8 

5.9 

6.0 

6.2 

6.3 

6.4 

6.6 

6.7 

7.4 

8.0 

8.7 

9.4 

10.0 

13.4 

5.9 

6.0 

6.1 

6.3 

6.4 

6.6 

6.7 

6.8 

7.0 

7.7 

8.4 

9.1 

9.8 

10.5 

14  0 

6.1 

U.3 

6.4 

6.5 

6.7 

6.8 

7.0 

7.1 

7.3 

8.0 

8.7 

9.5 

10.2 

10.9 

14  5 

6.4 

6.5 

6.7 

6.8 

7.0 

7.1 

7.3 

7.4 

7.6 

8.3 

9.1 

9.8 

10.5 

11.3 

15.1 

6.6 

6.8 

6.9 

7.1 

7.2 

7.4 

7.5 

7.7 

7.9 

8.6 

9.4 

10.2 

11.0 

11.8 

15.7 

6.8 

7.0 

7.2 

7.3 

7.5 

7.7 

7.8 

8.0 

8.1 

9.0 

9.7 

10.6 

11.4 

12.2 

16.3 

7.1 

7.3 

7.4 

7.6 

7.8 

7.9 

8.1 

8.3 

8.4 

9.3 

10.1 

11.0 

11.8 

12.7 

16.9 

7.3 

7.5 

7.7 

7.9 

8.0 

8.2 

8.4 

8.6 

8.7 

9.6 

10.5 

11.3 

12.2 

13.1 

17.5 

8.6 

8.8 

9.0 

9.2 

9.4 

9.6 

9.8 

10.0 

10.2 

11.2 

12.2 

13.2 

14.3 

15.3 

20.4 

9.8 

10.0 

10.2 

10.5 

10.6 

10.9 

11.2 

11.4 

11.6 

12.8 

U.O 

15.1 

16.3 

17.4 

23.3 

11.0 

11.3 

11.5 

11.8 

12.0 

12.3 

12.6 

12.8 

13.1 

14.4 

15.7 

17.0 

18.3 

19.6 

26.2 

12.2 

12.5 

12.8 

13.1 

13.4 

13.7 

14  0 

14.2 

14.5 

16.0 

17.4 

18.9 

20.3 

21.8 

29.1 

13.4 

13.8 

14.1 

14  4 

14  7 

15.0 

15.4 

15.7 

16.0 

17.6 

19.2 

20.8 

22.4 

24  0 

32.0 

14.7 

15.0 

15.4 

15.7 

16.1 

16.4 

16.8 

17.1 

17.5 

19.2 

20.9 

22.7 

24.4 

26.2 

34  9 

17.1 

17.5 

17.9 

18.3 

18.7 

19.1   • 

19.5 

20.0 

20.4 

22.4 

24  4 

26.5 

28.5 

30.5 

40.7 

19.5 

20.0 

20.5 

20.9 

21.4 

21.9 

22.3 

22.8 

23.3 

25^6 

27.9 

30.2 

32.6 

34  9 

40.5 

22.0 

22.5 

23.0 

23.6 

24.1 

24.6 

25.1 

25.6 

26.2 

28.8 

31.4 

34.0 

36.6 

39.3 

52.3 

24.4 

25.0 

25.6 

26.2 

26:7 

27.3 

27.9 

28.5 

29.1   . 

32.0 

34.9 

37.8 

40.7 

43.6 

58.1 

26.9 

27.5 

28.1 

28.8 

29.4 

30.1 

30.7 

31.3 

32.0 

35.2 

38.4 

41.6 

41.8 

48.0 

64.0 

29.3 

30.0  ■ 

30.7 

31.4 

32.1 

32.8 

33.5 

34.2 

34.9 

38.4 

41.9 

45.3 

48.8 

52.3 

69.8 

31.7 

32.5 

33.2 

34.0 

34.8 

35.5 

36.3 

37.0 

37.8 

41.6 

45.3 

49.1 

52.9 

56.7 

75.6 

34  2 

35.0 

35.8 

36,6 

37.4 

38.2 

39.1 

39.9 

40.7 

44  7 

48.8 

.52  9 

57.0 

61.0 

81.4 

36.6 

37.5 

38.4 

39.2 

40.1 

41.0 

41.8 

42.7 

43.6 

47.9 

52.3 

56.7 

61.0 

6.5.4 

87.2 

39.0 

40.0 

40.9 

41.8 

42.8 

43.7 

44  6 

45.6 

46.5 

51.1 

55.8 

60.4 

65.1 

69.7 

93.0 

41.5 

42.5 

43.4 

44.4 

45.4 

46.4 

47.4 

48.4 

49.4 

54.3 

59.2 

64.2 

69.1 

74.1 

98.7 

43.9 

44  9 

46.0 

47.0 

48.1 

49.1 

50.2 

51.2 

52.3 

57.5 

62.7 

67.9 

73.2 

78.4 

104  5 

58.5 

59.8 

61.2 

62.6 

64  0 

65.4 

66.8 

68.2 

69.6 

76.5 

83.5 

90.5 

97.4 

104.4 

139.2 

72.9 

74.7 

76.4 

78.1 

79.9 

81.6 

83.3 

85.1 

86.8 

95.5 

104  2 

112.9 

121.5 

130.2 

173.6 

D 

D 

D 

D 

D 

D 

D 

D 

D 

I) 

D 

D 

D 

D 

D 

840 

860 

880 

900 

920 

940 

960 

980 

1,000 

1,100 

1,200 

1,300 

1,400 

1,500 

2,000 

160 


A  MANCTAL  OF  TOPOGRAPHIC  METHODS. 


Computed  from  the  formula  k='D  sin  a  ( 


Table  XI. — Differences  of  altitude 

[Prepared  by 

1  a,  in  ■whicli  D  is  tlie  observed  distance  of  the 


D 

D 

D 

D 

D 

D 

D 

» 

D 

D 

I) 

D 

D 

D 

" 

100 

;iio 

120 

130 

140 

150 

160 

170 

180 

190 

200 

220 

240 

260 

0    01 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0    02 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

-    0.1 

0.2 

0    03 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0    0-1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0  05 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0   06 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

0   07 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0  08 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0,5 

0.5 

0.6 

0.6 

0   09 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

0.6 

0.7 

0  10 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0,6 

0.6 

0.6 

0.7 

0.8 

0    11 

0.3 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.8 

0.8 

0  12 

0.3 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0   13 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

0   U 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

1.1 

0  15 

0.4 

0.5 

0.5 

0.6 

6.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

1.0 

1.1 

0    16 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

0    17 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

3.3 

0    18 

0.5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.2 

1.3 

1.4 

0    19 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.1 

1.1 

1.2 

1.3 

1.4 

0  20 

0.6 

0.6 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

0  21 

0.6 

0.7 

0.7 

0.8 

0.9 

0.9 

1.0 

l.O 

1.1 

1.2 

1.2 

1.3 

1.5 

1.6 

0  22 

0.6 

0.7 

0.8 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

1.4 

1.5 

1.7 

0  23 

0.7 

0,7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.1 

1.2 

1.3 

1.3 

1.5 

1.6 

1.7 

0  24 

0.7 

0.8 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.3 

1.4 

1.5 

1.7 

1.8 

0  25 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.9 

0  26 

0.8 

0.8 

0.9 

1.0 

1.1 

1.1 

1.2 

1.3 

1.4 

1.4 

1.5 

1.7 

1.8 

2.0 

0   27 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1.3 

1.3 

1.4 

1.5 

1.6 

1.7 

1.9 

2.0 

0   28 

0.8 

0.9 

1.0 

1.1 

1.1 

1.2 

1.3 

1.4 

1.5 

1.5 

1.6 

1.S 

2.0 

2.1 

0   29 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.4 

1.5 

1.6 

1.7 

1.9 

2.0 

2.2 

0  30 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.7 

1.9 

2.1 

2.3 

0   35 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.2 

2.4 

2.6 

0    40 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.9 

2.0 

2.1 

2.2 

2.3 

2.6. 

2.8 

3.0 

0    43 

1.3 

1.4 

1.6 

1.7 

1.8 

2.0 

2.1 

2.2 

2.4 

2.5 

2.6 

2.9 

3.1 

3.4 

0    50 

1.5 

1.6 

1.7 

1.9 

2.0 

2.2 

2.3 

2.5 

2.6 

2.8 

2.9 

3. -2 

■3.5 

3.8 

0   55 

1.6 

1.8 

1.9 

2.1 

2.2 

2.4 

2.6 

2.7 

2.9 

3.0 

3.2 

3.5 

3.8 

4.2 

1  00 

1.7 

1.9 

2.1 

2.3 

2.4 

2.6 

2.8 

3.0 

3.1 

3.3 

3.5 

3.8 

4.2 

4.5 

1    10 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.3 

3.5 

3.7 

3.9 

4.1 

4.5 

4.9 

6.3 

1  20 

2.3 

2.6 

2.8 

3.0 

3.3 

3.5 

3.7 

4.0 

4.2 

4.4 

4.7 

5.1 

6.6 

6.0 

1   30 

2.6 

2.9 

3.1 

3.4 

3.7 

3.9 

4.2 

4.4 

4.7 

5.0 

5.2 

5.8 

6.3 

6.8 

1    40 

2.9 

3.2 

3.5 

3.8 

4.1 

4.4 

4.7 

4.9 

5.2 

5.5 

5.8 

6.4 

7.0 

7.6 

1   50 

3.2 

3.5 

3.8 

4.2 

4.5 

4.8 

5.1 

5.4 

5.8 

6.1 

6.4 

7.0 

7.7 

8.3 

2  00 

3.5 

3.8 

4.2 

4.5 

4.9 

5.2 

5.6 

5.9 

6.3 

6.6 

7.0 

7.7 

8.4 

9.1 

2   10 

3.8 

4.2 

4.5 

4.9 

5.3 

5.7 

6.0 

6.4. 

6.8 

7.2 

7.6 

8.3 

9.1 

9.8 

2   20 

4.1 

4.5 

4.9 

5.3 

5.7 

6.1 

6.5 

6.9 

7.3 

7.7 

8.1 

8.9 

9.8 

10.  a 

2   30 

4.4 

4.8 

5.2 

5.7 

6.1 

6.5 

7.0 

7.4 

7.8 

8.3 

8.7 

9.6 

10.5 

11.3 

2  40 

4.6 

6.1 

5.6 

6.0 

6.5 

7.0 

7.4 

7.9 

8.4 

8.8 

9.3 

10.2 

11.2 

12.1 

2  50 

4.9 

5.4 

5.9 

6.4 

6.9 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.9 

11.8 

12.8 

300 

5.2 

5.7 

6.3 

6.8 

7.3 

7.8 

8.4 

8.9 

9.4 

11.9 

10.5 

11.5 

12.5 

13.6 

4  00 

7.0 

7.7 

8.4 

9.0 

9.7 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

15.3 

16.7 

18.1 

3  00 

8.7 

9.6 

10.4 

11.3 

12.2 

J3.0 

13.9 

14.8 

15.6 

16.5 

17.4 

19.1 

20.8 

22.6 

D 

D 

D 

D 

D 

D 

D 

D 

D 

» 

D 

D 

D 

D 

" 

100 

110 

120 

130 

140 

150 

160 

170 

180 

190 

200 

220 

240 

260 

ALTITUDE  TABLES. 


161 


from  telemeter  measures — Continued. 

E.  S.  ■Woodward.] 

telemeter  staff,  a  13  the  angle  of  elevation  or  depression,  and  h  is  the  difference  in  height. 


D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

280 

300 

320 

840 

360 

380 

400 

420 

410 

460 

480 

500 

520 

540 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.6 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0.6 

0.6 

0.7 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

0.9 

1.0 

1.0 

1.1 

1.1 

0  7 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.1 

1.2 

1.2 

1.3 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

1.3 

1.4 

1.4 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

1.3 

1.4 

1.5 

1.5 

1.6 

0.9 

1.0 

1.0 

1.1 

1.2 

1.2 

1.3 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.7 

1.0 

1.0 

1.1 

1.2 

1.3 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.7 

1.8 

1.9 

1.1 

1.1 

1.2 

1.3 

1.4 

1.4 

1.5 

1.6 

1.7 

1.7 

1.8 

1.9 

2.0 

2.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.0 

2.1 

2.2 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

1.3 

.1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.0 

'  2.1 

2.2 

2.3 

2.4 

2.5 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

2.8 

1.5 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.5 

2.7 

2.8 

2.9 

3.0 

1.6 

1.7 

1.9 

2.0 

2.1 

2.2 

2.3 

2.4 

2.6 

2.7 

2.8 

2.9 

3.0 

3.1 

1.7 

1.8 

2.0 

2.1 

2.2 

2.3 

2.4 

2.6 

2.7 

2.8 

2.9 

3.1 

3.2 

3.3 

1.8 

1.9 

2.0 

2.2 

2.3 

2.4 

2.6 

2.7 

2.8 

2.9 

3.1 

3.2 

3.3 

3.5 

1.9 

2.0 

2.1 

2.3 

2.4 

2.5 

2.7 

2.8 

2.9 

3.1 

3.2 

3.3 

3.5 

3.6 

2.0 

2.1 

2.2 

2.4 

2.5 

2.7 

2.8 

2.9 

3.1 

3.2 

3.4 

3.5 

3.6 

3.8 

2.0 

2.2 

2.3 

2.5 

2.6 

2.8 

2.9 

3.1 

3.2 

3.3 

3.5 

3.6 

3.8 

3.9 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.0 

3.2 

3.3 

3.5 

3.6 

3.8 

3.9 

4.1 

2.2 

2.4 

2.5 

2.7 

2.8 

3.0 

3.1 

3.3 

3.5 

3.6 

3.8 

3.9 

4.1 

4.2 

2.3 

2.4 

2.6 

2.8 

2.9 

3.1 

2.3 

3.4 

3.6 

3.7 

3.9 

4.1 

4.2 

4.4 

2.4 

2.5 

2.7 

2.9 

3.0 

3.2 

3.4 

3.5 

3.7 

3.9 

4.1 

4.2 

4.4 

4.6 

2.4 

2.6 

2.8 

3.0 

3.1 

3.3 

3.5 

3.7 

3.8 

4.0 

4.2 

4.4 

4.5 

4.7 

S.9 

3.1 

3.3 

3.5 

3.7 

3.9 

4.1 

4.3 

4.5 

.  4.7 

4.9 

5.1 

5.3 

5.5 

3.3 

3.5 

3.7 

4.0 

4.2 

•  4.4 

4.7 

4.9 

5.1 

5.3' 

5.6 

5.8 

6.0 

6.3 

3.7 

3.9 

4.2 

4.5 

4.7 

5.0 

5.2 

5.5 

5.8 

6.0 

6.3 

6.5 

6.8 

7.1 

4.1 

4.4 

4.7 

4.9 

5.2 

5.5 

5.8 

6.1 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

4.5 

4.8 

5.1 

5.4 

5.8 

6.1 

6.4 

6.7 

7.0 

7.4 

7.7 

8.0 

8.3 

8.6 

4.9 

5.2 

6.6 

5.9 

6.3 

6.6 

7.0 

7.3 

7.7 

8.0 

8.4 

8.7 

9.1 

9.4 

5.7 

6.1 

6.5 

6.9 

7.3 

7.7 

8.1 

8.6 

9.0 

9.4 

9.8 

10.2 

10.6 

11.0 

6.5 

7.0 

7.4 

7.9 

8.4 

8.8 

9.3 

9.8 

10.2 

10.7 

11.2 

11.6 

12.1 

12.6 

7.3 

7.9 

8.4 

8.9 

9.4 

9.9 

10.5 

11.0 

11.5 

12.0 

12.6 

13.1 

13.6 

14.1 

8  1 

8.7 

9.3 

9.9 

10.5 

11.0 

11.6 

12.2 

12.8 

13.4 

14.0 

14.5 

15.1 

15.7 

9.0 

9.6 

10.2 

10.9 

11.5 

12.2 

12.8 

13.4 

14.1 

14.7 

15.4 

16.0 

16.6 

17.3 

9.8 

10.5 

11.2 

11.9 

12.6 

13.3 

14.0 

14.6 

15.3 

16.0 

16.7 

17.4 

18.1 

18.8 

10.6 

11.3 

12.1 

12.8 

13.6 

14.4 

16.1 

15.9 

16.6 

17.4 

18.1 

18.9 

19.6 

20.4 

11.4 

12.2 

13.0 

13.8 

14.6 

15.5 

16.3 

17.1 

17.9 

18.7 

19.5 

20.3 

21.2 

22.0 

12.2 

13.1 

13.9 

14.8 

15.7 

16.6 

17.4 

18.3 

19.2 

20.0 

20.9 

21.8 

22.7 

23.5 

13.0 

13.9 

14.8 

15.8 

16.7 

17.7 

18.6 

19.5 

20.5 

21.4 

22.3 

2S.2 

24.2 

25.1 

13.8 

14.8 

15.8 

16.8 

17.8 

18.8 

19.7 

20.7 

21.7 

22.7 

23.7 

24.7 

25.7 

26.7 

14.6 

15.7 

16.7 

17.8 

18.8 

19.9 

20.9 

21.9 

23.0 

24.0 

25.1 

26.1 

27.2 

28.2 

19.5 

20.9 

22.3 

23.7 

25.1 

26.4 

27.8 

29.2 

30.6 

32.0 

33.4 

34.8 

36.2 

37.6 

24.3 

26.0 

27.8 

29.5 

31.3 

33.0 

34.7 

36.5 

38.2 

39.9 

41.7 

43.4 

45.1 

46.9 

D 

D 

D 

I) 

D 

D 

D 

D 

D 

D 

D 

D 

D 

D 

280 

300 

320 

340 

360 

380 

400 

420 

440 

460 

480 

600 

620 

540 

-11 


162 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


CONSTANTS. 


163 


Table  XIII.— Constants. 

IT =3.141593 

log.ir =0.4971499 

180°  1 

"ir^arc  lo— 57°.29578=57°  17'  44" .8;  log.  =1.7581226 

^i5?52!=_J_=3447' .74677:  log.=3. 5362739 
n        arc  1'  " 

r"=ii?552^^i— =206264".80625:  loe.=5.  314(251 
IT  Sin  1"  ^ 

comp .  log.  =4 .  6855749 
=log.  sin  1" 
Log. 

Number  of  degrees  in  circumference 360=2. 5563025 

Number  of  minutes  in  circumference 21,600=4.3344538 

Number  of  seconds  in  circumference 1, 296, 000=6. 1126050 

Lengtli  of  arc  of  1  degree 0174,5329=8.  2418774—10 

Lengtb  of  arc  of  1  minute 00029089=6. 4637261—10 

Lengtli  of  arc  of  1  second 000004848=4. 6855749—10 

Constants  of  generating  ellipse  of  Clarke's  spheroid. 


e'=  (l—  *,  ^    =0.  00676866 
»=(1— VlIIj2)(l+<7i:ii2)-i=  O.C 


7.  8305030—10 
7.  2299162—10 

Length  of  the  meter  in  inches  according  to  various  authoriiiei. 

Inches. 
1  meter=39.  370432,  Clarke,  1866-1873. 
=39.  370790,  Kater,  1818. 

=39.368505,  Coast  Survey,  1851-1858  (Hassler  corrected). 
=39. 38092,    Hassler,  1832. 
=39. 36985,    L.ake  Survey,  1885. 

=39.377786,  Theoretical  ten-millionth  of  quadrant  (Clarke). 
=39. 37,         By  act  of  Congress,  1866. 

The  standard  meter  has  its  normal  length  at  32'^  E 
The  standard  yard  has  its  normal  length  at  62°  F. 
The  value  first  given  is  the  one  generally  adopted  by  scientific  men  in 
the  United  States. 

Values  adopted  in  the  measurement  of  an  arc  of  parallel  extending  from  Ireland  to  the  river  Ural  in  Russia, 
as  the  exact  relative  lengths  of  standards  used  as  the  units  of  measure  in  the  triangulations  of  England, 
France,  Belgium,  Prussia,  and  Russia. 


Standards. 

Expressed  in 
terms  of  the 
standard  yard. 

Expressed  in 
inches. 

Expressed  in 

lines  of  the 

toise. 

Expressed  in 
millimeters. 

1. 00000000 
2. 13151116 
1. 09362311 

36.000000 
76.  734402 
39.  370432 

405.  34622 
864.00000 
443.29600 

914. 39180 
1, 949. 03632 
1, 000. 00000 

CONVERSION  TABLES. 

Table  XIV. — Meters  into  yards. 

[Extracted  from  Appendix  No.  6,  TJ.  S.  Coast  and  Geodetic  Survey  Eeport  for  1884]. 

[1  meter  =  1.093623  yards.] 


Meters. 

Yards. 

Meters. 

Yards. 

Meters. 

Yards. 

Meters. 

Yards. 

Meters. 

Yards. 

100,000 

109,  362.  3 

90, 000 

98,426.1 

9,000 

9,  842.  61 

900 

984.26 

90 

98. 426 

9 

9.843 

80,  000 

87,489.8 

.    8,000 

8,  748.  98 

800 

874. 90 

80 

87.  490 

8 

8.749 

70,  000 

76, 553.  6 

7,000 

7,  655.  36 

700 

765.  54 

70 

76.  554 

7 

7.655 

60,  000 

65,617.4 

6,000 

6,  561.  74 

600 

656. 17 

60 

65.  617 

6 

6.562 

50, 000 

54,681.2 

5,000 

5,  468. 12 

500 

546.81 

50 

54.  681 

5 

5.568 

40,  000 

43,  744.  9 

4,000 

4,  374.  49 

400 

437.  45 

40 

43.745 

4 

4.374 

30,  000 

32,  808.  7 

3,000 

3,280.87 

300 

328.  09 

30 

32.  809 

3 

3.281 

20,  000 

21,  872.  5 

2,000 

2, 187.  25 

200 

218.72 

20 

21.  872 

2 

2.187 

10,  000 

10,  936. 2 

1,000 

1,  093.  62 

100 

109.  36 

10 

10.  936 

1 

1.094 

164 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XV. — Yards  into  meters. 
[1  yard  =  0.914392  meter.] 


Tarda. 

Meters. 

Yards. 

Meters. 

Yards. 

Meters. 

Yards. 

Meters. 

Yards. 

Meters. 

100, 000 

91,439.2 

90,  000 

82,295.3 

9,000 

8, 229. 53 

900 

822.95 

90 

82.295 

9 

8.230 

SO,  000 

73, 151.  3 

8,000 

7,  315. 13 

800 

731.  51 

80 

73. 151 

8 

7.315 

70,  000 

61,  007. 4 

7,000 

6,  400.  74 

700 

640.07 

70 

64.  007 

7 

6.401 

60,  OOO 

54,863.5 

6,000 

5,486.35 

600 

548.  64 

5.486 

50, 000 

45,  719.  6 

5,000 

4,  571. 96 

500 

457.20 

50 

45.  720 

5 

4.572 

40,  000 

36,  575. 7 

4,000 

3,  657.  57 

400 

365.  76 

40 

36.  576 

4 

3.658 

30,  000 

27,431.8 

3,000 

2,  743. 18 

300 

274.  32 

3 

2.743 

20,  000 

18,287.8 

2,000 

1,  828.  78 

200 

182. 88 

20 

18.  288 

2 

1.829 

10,  000 

9, 143. 9 

1,000 

914.39 

100 

91.44 

10 

9.144 

1 

0.914 

Table  XVI. — Meters  into  inches  and  inches  into  meters. 
[1  meter  =  39.370432  inches,    log.  =  1.5951702.]  [1  iucli  =  0.02539977  meter,    log.  =8.4048298.] 


Meters. 

Inches. 

1 

39.37043 

.      2 

78.  74086 

3 

118.11130 

4 

157.  48173 

5 

196.  85216 

6 

236.  22259 

7 

275.  59302 

8 

314.  96346 

9 

354. 33389 

Inches. 

Meters. 

1 

0.  025400 

2 

0.  050800 

3 

0.  076199 

4 

0. 101599 

5 

0. 126999 

6 

0. 152399 

7 

0. 177798 

8 

0. 203198 

9 

0.  228598 

Table  XVII. — Meters  into  statute  and  nautical  miles. 


Meters. 

Statnte 
miles. 

Nautical 
miles. 

Meters. 

Statute 
miles. 

Nautical 
miles. 

Meters. 

Statnte 
miles. 

Nautical 
miles. 

Meters. 

Statute 
miles. 

Nautical 
miles. 

100, 000 

62. 138 

53.959 

90, 000 

55.924 

48. 563 

9,000 

5.592 

4.856 

900 

0.559 

0.486 

90 

0.056 

0.049 

80, 000 

49.  710 

43. 167 

8,000 

4.971 

4.317 

800 

0.497 

0.432 

80 

0.  '50 

70,  000 

43.496 

37.  772 

7,000 

4.350 

3.777 

700 

0.435 

0.378 

70 

0.043 

0.038 

60,  000 

37.  283 

32.  376 

6,000 

3.728 

3.238 

600 

0.373 

0.324 

60 

0.037 

0.032 

50,  000 

31.  069 

26.  980 

5,000 

3.107 

2.698 

500 

0.311 

0.270 

50 

0.031 

0.027 

40,  000 

24.855 

21.  584 

4,000 

2.486 

2.158 

400 

0.249 

0.216 

40 

0.025 

0.022 

30,  000 

18.641 

16.188 

3,000 

1.864 

1.619 

300 

0.186 

0.162 

30 

0.019 

0.016 

20,  000 

12.428 

10.  792 

2,000 

1.243 

1.079 

200 

0.124 

O.108 

20 

0.012 

0.011 

10,  000 

6.214 

5.396 

1,000 

0.621 

0.540 

100 

0.062 

0.054 

10 

0.006 

0.005 

Table  XVIII. — Statute  and  nautical  miles  into  meters. 


Meters  in  [Meters  in 

Meters  in 

Meters  in 

Meters  in 

Meters  in 

Miles. 

Miles. 

statute 

nautical 

Miles. 

statute 

nautical 

Miles. 

statute 

nautical 

miles. 

miles. 

miles. 

miles. 

miles. 

miles. 

miles. 

miles. 

100 

160, 933. 0 

185,324.8 

90 

144,839.7 

166.  792.  3 

9 

14,483.97 

16, 679. 23 

.9 

1,448.40 

1,  667.  92 

.09 

144.84 

166. 79 

80 

128.  746.  4 

148,  259. 8 

8 

12,874.64 

14,  825.  98 

.8 

1,287.46 

1,  482.  60 

70 

112,  653. 1 

129,  727. 4 

7 

11,  265.  31 

12,  972. 74 

.7 

1, 126.  53 

1,297.27 

.07 

112.  65 

129. 73 

60 

96,  559.  8 

111,  194.  9 

6 

9,  655.  98 

11,119.49 

.6 

965.  60 

1,  111. 95 

.06 

96.56 

111.  19 

50 

80,466.5 

92,662.4 

5 

8,046.65 

9,  266.  24 

.5 

804.67 

926.  62 

.05 

40 

64,373.2 

74, 129.  9 

4 

6,437.32 

7,  412.  99 

.4 

643. 73 

741.  30 

30 

48,  279. 9 

65,  597.  4 

3 

4,827.99 

5,  559. 74 

.3 

482.  80 

20 

32,186.6 

37,  065.  0 

2 

3,  218.  66 

3,706.50 

.2 

321.  87 

370.  65 

.02 

32.19 

37.06 

10 

16,093.3 

18,532.5 

1 

1,609.33 

1,  853.  25 

.1 

160.  93 

185.  32 

.01 

16.09 

18.53 

IMeters  x  39.370432 
Meters  x  3.280869 
Meters  x  1.093623 
Meters  X    0.000621377  : 


:iiiclies,  or  to  log.  of  meters  add  1.5951701 
:  feet,  or  to  log.  of  meters  add  0.5159889 
:  yards,  or  to  log.  of  meters  add  0.0388676 
:  miles,   or  to  log.  of  meters  add  6.7933550 


PROJECTION  TABLES. 


165 


Table  XIX. — For  projection  of  maps  of  large  areas. 
[Extracted  from  Appendix  'So.  6,  TJ.  S.  Coast  and  Geodetic  Survey  Report  for  1884.] 

LENGTHS  OF  DEGREES  OF  THE  MERIDIAN. 


Latitude 

Meters.* 

Statute  miles. 

Latitude 

Meters.* 

Statute  miles. 

0 

110,567.2 

68. 704 

45 

Ill,  130. 9 

69.064 

1 

110,  567. 6 

68.  704 

46 

111,  150. 6 

69.  066 

2 

110,  568. 6 

68.  705 

47 

111,  170.  4 

69.  079 

3 

110,  570.  3 

68.  706 

48 

111,  190. 1 

69.  091 

4 

110,  572.  7 

68.  708 

49 

111,  209.  7 

69. 103 

5 

110,  675.  8 

68.  710 

50 

111,229.3 

69. 115 

6 

110,  579.  5 

68.  712 

51 

111,248.7 

69. 127 

7 

110,  583.  9 

68,715 

52 

111,  208. 0 

69. 139 

8 

HO,  589.  0 

68.  718 

63 

HI,  287. 1 

69. 151 

9 

110,  594.  7 

68.  721 

54 

111,  306.  0 

69. 163 

10 

110,  601. 1 

68.  725 

65 

111,324.8 

69,  175 

11 

110,  608. 1 

68.  730 

56 

111,  343.  3 

69. 186 

12 

110,  615.  8 

68.  734 

57 

111,  361.  5 

69. 197 

13 

110,  624. 1 

68.  739 

58 

HI,  379.  5 

69. 209 

14 

110,  633.  0 

68.744 

59 

111,  397.  2 

69.  220 

15 

110,042.5 

63. 751 

60 

111,414.5 

69.  230 

16 

110,  652.  6 

68.  757 

61 

111,431.5 

69.  241 

17 

110,  663.  3 

68.  764 

62 

111,  448. 2 

69.  251 

18 

110,674.5 

68.  771 

63 

111,  464.  4 

69. 261 

19 

110,  686.  3 

68.  778 

64 

111,  480.  3 

69.271 

20 

110,  698.  7 

68.  786 

65 

111,  495.  7 

69. 281 

21 

110,  711. 6 

68.  794 

66 

111,510.7 

69.  290 

22 

HO,  725.  0 

68.  802 

67 

111,  525.  3 

69.  299 

23 

HO,  738.  8 

68.811 

68 

111,  539.  3 

69.  308 

24 

110,  753.  2 

68.  820 

69 

111,552.9 

69.  316 

25 

110,  768.  0 

68.829- 

70 

111,565.9 

69.  324 

26 

110,  783. 3  • 

68.  839 

71 

111,578.4 

69.  332 

27 

110,799.0 

68.  848 

72 

111,590.4 

69.  340 

28 

119,  815. 1 

68.  858 

73 

HI,  601.  8 

69.  347 

29 

110,  831.  6 

68.  869 

74 

111,612.7 

69.  354 

30 

110,848.5 

68.  879 

75 

111,  622.  9 

69.  360 

31 

110,  865.  7 

68. 890 

76 

111,  632.  6 

69.  366 

32 

110,  883. 2 

68.  91)1 

77 

111,  841. 6 

69. 372 

33 

110,  901. 1 

68.  912 

78 

111,650.0 

69.  377 

34 

110,  919.  2 

68.  923 

79 

111,  657.  8 

69.  382 

35 

110,  937.  6 

68.  935 

80 

111,  664.  9 

69.  386 

36 

110,  956.  2 

68. 946 

81 

111,671.4 

69.  390 

37 

110,  975. 1 

68. 958 

82 

111,  677.  2 

69. 394 

38 

110,  994. 1 

68.  969 

83 

111,  682.  4 

69.  397 

39 

111,013.3 

68.  981 

84 

111,  686.  9 

69. 400 

40 

111,032.7 

68. 993 

85 

111,  690.  7 

69. 402 

41 

111,  052.  2 

69.006 

86 

111,  693.  8 

69. 404 

42 

111,  071.  7 

69.  018 

87 

111,  696.  2 

69.  405 

43 

111,091.4 

•69.  030 

88 

HI,  697. 9 

69.407 

44 

111,  111.  1 

69.043 

89 

111,  699. 0 

69.407  . 

45 

111,130.9 

69.054 

90 

111,699.3 

69.407 

*  These  quantities  express  tlie  number  of  meters  and  statute  miles  contained  "within  an  arc  of  which  the  degree  of  lati- 
tude numed  is  the  middle;  thus,  the  quantity,  111032.7,  opposite  latitude  40°,  is  the  number  of  meters  between  latitude  39° 
30'  and  latitude  40°  30'. 


166 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XIX. — For  projection  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6,  U.  S.  Coast  and  Geodetic  Survey  Report  for  1884.] 


LENGTHS  OP  DEGREES  OF  THE  PARALLEL. 


Latitude. 

Meters. 

Statute  miles. 

Latitude. 

Meters. 

Statute  miles. 

0 

Ill,  321 

69. 172 

45 

78, 849 

48.995 

1 

1,304 

9.162 

46 

7,466 

8.136 

2 

1,253 

9.130 

47 

6,058 

7.261 

3 

1,169 

9.078 

48 

4,628 

6.372 

4 

1,051 

9.005 

49 

3,174 

5.469 

6 

110, 900 

68.  911 

50 

71,  698 

44,  552 

6 

0,715 

8.795 

51 

70,  200 

3.621 

7 

0,497 

8.660 

52 

68,  680 

2.676 

8 

0,245 

8.504 

53 

7,140 

1.719 

9 

109, 959 

8.326 

54 

5,578 

0.749 

10 

109, 641 

68. 129 

55 

63,  996 

39.  766 

U 

9,289 

7.910 

56 

2,395 

8.771 

12 

8,904 

7.670 

57 

60,  774 

7.764 

13 

8,486 

7.410 

58 

69, 135 

6.745 

14 

8,036 

7.131 

59 

7,478 

5.716 

15 

107,  553 

66.  830 

60 

55,  802 

34.  674 

16 

7,  036 

6.510 

61 

4,110 

3.623 

17 

6,487 

6.169 

62 

2,400 

2.560 

18 

5,906 

5.808 

63 

50,  675 

1.488 

19 

5,294 

5.427 

64 

48,  934 

0.406 

20 

104,  649 

65.  026 

65 

47, 177 

29.  315 

21 

3,972 

4.606 

66 

5,  407 

8.215 

22 

3.264 

4.166 

67 

3,622 

7.106 

23 

2,524 

3.706 

68 

1,823 

5.988 

24 

1,754 

3.228 

69 

0,012 

4.862 

25 

100,  952 

62. 729 

70 

38, 188 

23.  729 

26 

100. 119 

2.212 

71 

6,353  • 

2.589 

27 

99,  257 

1.676 

72 

4,606 

L441 

28 

8,364 

1.122 

73 

2,648 

20.  287 

29 

7,441 

0.548 

74 

0,781 

19. 127 

30 

96,488 

59.  956 

75 

28,  903 

17.  960 

31 

5,506 

9.345 

76 

7,017 

6.788 

32 

4,495 

8.716 

77 

5,123 

5.611 

33 

3,455 

8,071 

78 

3,220 

4.428 

34 

2,387     ■ 

7.407 

79 

1,311 

13.  242 

35 

91,  290 

56.  725 

80 

19,  394 

12.  051 

36 

90, 166 

6.027 

81 

17,  472 

10.  857 

37 

89,  014 

5.311 

'  82 

15,  545 

9.659 

38 

7,835 

4.579 

83 

13,  612 

8.458 

39 

6,629 

3.829 

84 

11,  675 

7.255 

40 

85,  396 

53.  063 

85 

9,735 

6.049 

41 

4,137 

2.281 

86 

7,792 

4.842 

42 

'2,  853 

1.483 

87 

5,846 

3.632 

43 

1,543 

50.  669 

88 

3,898 

2. 422     . 

44 

80,  208 

49.  840 

89 

1,  949 

1.211 

45 

78,  849 

48. 995 

90 

" 

0.000 

PBOJECTION  TABLES. 


167 


Table  XIX. — For  projection  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6,  TJ.  S.  Coast  and  Geodetic  Survey  Eeport  for  1884.] 


AECS  OF  THE  PARALLEL  IN  METERS. 


Latitude. 

Value  of  1'. 

Latitude. 

Value  of  1'. 

Latitude. 

Value  of  1'. 

24    00 

1695.9 

33    00 

1557.  6 

42    00 

1380.  9 

10 

3.7 

10 

4.7 

10 

77.3 

20 

1.5 

20 

1.7 

20 

73.7 

30 

1689.3 

30 

48.7 

30 

70.0 

40 

7.0 

40 

5.8 

40 

66.4 

50 

4.8 

50 

2.8 

50 

62.7 

25    00 

1682.  5 

34    00 

1539.  8 

43    00 

1359. 1 

10 

80.3 

10 

6.8 

10 

55.4 

20 

1678.  0 

20 

3.7 

20 

51.7 

30 

5.7 

30 

0.7 

30 

48.0 

40 

3.3 

40 

27.6 

40 

44.3 

50 

1.0 

50 

4.6 

50 

40.5 

26    00 

1668.  7 

35    00 

1521.  5 

44    00 

1336. 8 

10 

6.3 

10 

18.4 

10 

33.1 

20 

3.9 

20 

15.3 

20 

29.3 

30 

1.5 

30 

12.2 

30 

25.5 

40 

1659. 1 

40 

09.1. 

40 

21.7 

50 

6.7 

50 

05.9 

50 

18.0 

27    00 

1654.  3 

36    00 

1502.  8 

45    00 

1314.  2 

10 

51.8 

10 

1499.  6 

10 

10.3 

20 

1649.4 

20 

6.4 

20 

06.5 

30 

6.9 

30 

3.2 

30 

02.7 

40 

4.4 

40 

0.0 

40 

1298.  8 

50 

1.9 

50 

86.8 

50 

95.0 

28    00 

1639. 4 

37     00 

1483.  6 

46    00 

1291.  0 

10 

6.9 

10 

80.3 

10 

87.2 

20      ' 

4.3 

20 

77.1 

20 

83.3 

30 

1.8 

30 

73.8 

30 

79.4 

40 

29.2 

40 

70.5 

40 

75.5 

50 

6.6 

50 

67.2 

50 

71.6 

29    00 

1624.0 

38    00 

1463.  9 

47    00 

1267.  6 

10 

21.4 

10 

60.6 

10 

63.7 

20 

18.8 

20 

57.3 

20 

59.7 

30 

6.1 

30 

53.9 

30 

55.8 

40 

3.5 

40 

50.6 

•       40 

51.8 

50 

0.8 

50 

47.2 

50 

47.8 

30    00 

1608. 1 

39    00 

1443.8 

48    00 

1243.8 

10 

5.4 

"10 

40.4 

10 

39.8 

20 

2.7 

20 

37.0 

20 

35.8 

30 

0.0 

30 

33.6 

30 

31.7 

40 

1597.  3 

40 

30.2 

40 

27.7 

50 

4.5 

50 

26.7 

50 

23.6 

31    00 

1591. 8 

40    00 

1423.3 

49    00 

1219.  6 

10 

89.0 

10 

19.8 

10 

15.5 

20 

6.2 

20 

16.3 

20 

11.4 

30 

3.4 

30 

12.8 

30 

07.3 

40 

0.6 

40 

09.3 

40 

03.2 

50 

77.8 

50 

05.8 

50 

1199.1 

32    00 

1574.  9 

41     00 

1402.  3 

50    00 

1195.  0 

10 

72.1 

10 

1398.8 

10 

90.8 

20 

69.2 

20 

95.2 

20 

86.7 

30 

6.3 

30 

91.6 

30 

82.5 

40 

3.4 

40 

88.1 

40 

78.4 

50 

0.5 

50 

84.5 

50 

74.2 

168 


A  MANUAL  OF  TOPOGKAPHIC  METHODS. 


Table  XIX, — For  projections  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6,  V.  S.  Coast  and  Geodetic  Survey  Keport  for  1884.] 

COORDINATES  OF  CUBVATUKE. 


NATURAL  SCALE.-VAI,TIES  OF  X  AND  Y  IN  METERS. 

Latitude  24°. 

Latitude  25°. 

Latitude  26°. 

Latitude  27°. 

Longi- 
tude. 

X 

T 

Longi- 
tude. 

X 

T 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

1    00 

101,  753 

361 

1 

00 

100,  951 

372 

1 

00 

100, 118 

383 

1 

00 

99,  256 

393 

2    00 

203,  500 

1,445 

2 

00 

201,  896 

1,489 

2 

00 

200,  231 

1,532 

2 

00 

198,  505 

1,573 

3     00 

305,237 

3,250 

3 

00 

302,  831 

3,351 

3 

00 

300,  332 

3,447 

3 

00 

297,  742 

3,539 

4    00 

406,  9d9 

5,778 

4 

00 

403,  749 

5,957 

4 

00 

400,  416 

6,128 

4 

00 

396,  960 

6,291 

5     00 

508,  660 

9,028 

5 

00 

504,  645 

9,307 

5 

00 

500,  476 

9,574 

5 

00 

496, 154 

9,829 

6    00 

610,  336 

13,  001 

6 

00 

605,  514 

13,401 

6 

00 

600,  506 

13,  786 

6 

00 

595,  316 

14,  154  : 

7    00 

711,981 

17,  695 

7 

00 

706,  349 

18,  239 

7 

00 

700, 501 

18,  763 

7 

00 

694,  440 

19, 204 

8    00 

313,590 

23, 109 

8 

00 

807, 146 

23,  821 

8 

00 

800,456 

24, 505 

8 

00 

793,  522 

25,  159 

9    00 

915, 159 

29,  245 

9 

00 

907,  899 

30, 146 

9 

00 

900,  364 

31,  Oil 

9 

00 

892,  554 

31, 839 

10    00 

J,  016,  681 

36, 102 

10 

00 

1,  008,  603 

37,  215 

10 

00 

1,  000,  218 

38,  282 

10 

00 

991,  529 

39,  303 

11    00 

1, 118, 152 

43, 679 

11 

00 

1, 109,  252 

45,  026 

11 

00 

1, 100, 015 

46,  316 

11 

00 

1,  090,  442 

47,  551 

12    00 

1,  219,  566 

51,977 

00 

1, 209,  841 

53,  578 

12 

00 

1,  199,  747 

55, 114 

12 

00 

1, 189,  287 

56,  583 

13    00 

1,320,919 

60,  994 

13 

00 

I,  310,  364 

62,  873 

13 

00 

1,  299,  409 

64,  675 

13 

00 

1,  288, 1157 

66,  398 

14    00 

1,  422,  205 

70,  731 

14 

00 

1,  410,  815 

72,  909 

14 

00 

1,  398,  994 

74,  998 

14 

00 

1,  386,  746 

76,  995 

15    00 

1,  523, 420 

81, 186 

15 

00 

1,  511, 190 

83,  685 

15 

00 

1,498,498 

86,  082 

15 

00 

1,  485,  348 

88,  374 

16    00 

1,  624,  558 

92,  360 

16 

00 

1,  611,  483 

95,  202 

16 

00 

1,  597,  914 

97, 928 

16 

00 

1,  583,  857 

100,  534 

17     00 

1,  725,  614 

104,  251 

'17 

00 

1,  711,  688 

107,  458 

17 

00 

1,  697,  237 

110,  534 

17 

00 

1,  682,  267 

113,  474 

18     00 

1,  826,  583 

116,  859 

18 

00 

1,  811,  800 

120,  453 

18 

00 

1,  796,  460 

123,  899 

18 

00 

1,780,570 

127, 193 

19    00 

1, 927,  460 

130,  184 

19 

00 

1,  911,  813 

134, 186 

19 

00 

1,  895,  578 

138,  023 

19 

00 

1,  878,  762 

141,690 

20    00 

2,  028, 240 

144,  225 

20 

00 

2,  Oil,  722 

148,  656 

20 

00 

1,  994,  585 

152,  905 

20 

00 

1,  976,  836 

156,  966 

21     00 

2, 128,  918 

158,  981 

21 

00 

2,  111.  522 

163,  862 

21 

00 

2,093,475 

168,  544 

21 

00 

2,  074,  786 

173,  018 

22    00 

2,  229,  488 

174,  451 

22 

00 

2.  211,  207 

179,  805 

22 

00 

2, 192,  243 

184, 939 

22 

00 

2, 172, 606 

189.  845 

23    00 

2,  329,  946 

190,  634 

23 

00 

2,  310,  771 

196,482 

23 

00 

2,  290,  882 

202,  089 

23 

00 

2,  270,  289 

207,  447 

24    00 

2,  430,  287 

207,  530 

24 

00 

2,410,210 

213,  894 

24 

00 

2, 389, 387 

219,  993 

24 

00 

2,  367,  830 

225,  823 

25    00 

2,  530,  505 

225, 138 

25 

00 

2,  609,  518 

232,  038 

25 

00 

2,  487,  753 

238,  650 

25 

00 

2,  465,  222 

244,  970 

26    00 

2,  650,  596 

243,  458 

26 

00 

2,  608,  689 

250,  914 

26 

00 

2,  585, 973 

258,  061 

26 

00 

2,  562,  459 

264,  889 

27    00 

2,  720,  554 

262,  487 

27 

00 

2, 707,  718 

270,  521 

27 

00 

2,  684, 042 

278,  222 

27 

00 

2,  659,  535 

285,  677 

28    00 

2,  830,  374 

282,  225 

28 

00 

2.  806,  600 

290,  859 

28 

00 

2,  781,  953 

299, 132 

28 

00 

2,  756,  445 

307,  035 

29    00 

2,  930,  052 

302,  671 

29 

00 

2,  905,  329 

311,925 

29 

00 

2,  879,  702 

320,  788 

29 

00 

2,  853, 181 

329,  259 

30    00 

3,  029,  582 

323,  825 

30 

00 

3,  003,  900 

333,  718 

30 

00 

2,  977,  281 

343, 197 

30 

00 

2,  949,  739 

352,  249 

PEOJECTION  TABLES. 


169 


Table  XIX. — For  projections  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6,  U.  S.  Coast  and  Geodetic  Survey  Report  for  1884.] 

COOHDINATES  OF  CUKVATDEE. 


NATURAL  SCALE.— VALUES  OF  X  AND  Y  IN  METERS. 

Latitude  28 

Latitude  29°. 

Latitude  30°. 

Latitude  31°. 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Longi. 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

1    00 

98,  363 

403 

1    00 

97, 439 

412 

1 

00 

96,  487 

421 

1 

00 

95,  505 

429 

2    00 

196,  719 

1,612 

2    00 

194,  872 

1,649 

2 

00 

192,  967 

1,684 

2 

00 

191,  002 

1,717 

3     00 

295,  062 

3,627 

3     00 

292,  291 

3,710 

3 

00 

289,  433 

3,789 

3 

00 

286,  484 

3,863 

i    00 

393,  385 

6,447 

4     00 

389,  689 

6,695 

4 

00 

385, 875 

6,735 

4 

00 

381,  943 

6,867 

5    00 

49],  682 

10,  073 

5    00 

487,  059 

10,  305 

5 

00 

482. 288 

10, 523 

5 

00 

477,  371 

10,  729 

6    00 

589,  945 

14,  505 

6    00 

584, 394 

14,  838 

6 

00 

578,  665 

15, 153 

6 

00 

572,  760 

15,450 

7    00 

688, 168 

19,  741 

7    00 

681,  687 

20, 194 

7 

00 

674,  998 

20,  623 

7 

00 

668, 103 

21,  027 

8    00 

786,  347 

25,  782 

8    00 

778,  931 

26, 374 

8 

00 

771,  279 

26,  934 

8 

00 

763,  392 

27,461 

9    00 

884,  472 

32,  627 

9    00 

876, 120 

33,  376 

9 

00 

867,  602 

34,  084 

9 

00 

858,  619 

34,  751 

10    00 

982,  537 

40,  276 

10     00 

973,  246 

41, 199 

10 

00 

963,  658 

43,  074 

10 

00 

953,777 

42,  897 

11    00 

1,  080,  637 

48,  728 

11    00 

1,  070, 302 

49, 845 

11 

00 

1,  059,  741 

50,  903 

11 

00 

1,  048,  858 

51,  898 

12    00 

1, 178,  464 

67, 983 

12    00 

1, 167,  282 

69,  313 

12 

00 

1, 165,  744 

60,  570 

12 

00 

1, 143,  854 

61,  753 

13     00 

1, 276,  312 

68,  040 

13     00 

1,  264, 178 

69,  601 

13 

00 

1,  251,  668 

71,  074 

13 

00 

1,  238,  758 

73,  462 

14    00 

1,374,075 

78,  699 

14     00 

1, 360,  983 

80.706 

14 

00 

1,  347,  477 

82,415 

14 

00 

1,  333,  561 

84,  024 

15    00 

1,  471,  745 

90,  558 

15    00 

1,457,691 

9%  631 

15 

00 

1,  443, 193 

94,  591 

15 

00 

1,  428,  267 

96,  437 

16    00 

1,  569,  315 

103,017 

16    00 

1,  554,  296 

105,  375 

16 

00 

1,  638, 800 

107,  603 

16 

00 

1,  522, 837 

109, 701 

17     00 

1,  666,  781 

116,  276 

17    00 

1,  650,  787 

118,  935 

17 

00 

1,  634,  290 

121,  449 

17 

00 

1,  617,  294 

133,  815 

18    00 

1,  764, 135 

130,  331 

18     00 

1,  747, 161 

133,311 

18 

00 

1,  729,  654 

136, 127 

18 

00 

1,  711,  621 

138,  777 

19    00 

1,  861,  371 

145, 185 

19    00 

1,  843,  410 

148,  502 

19 

00 

1,  824,  887 

151,  637 

19 

00 

1,805,810 

1.54,  586 

20    00 

1,958,481 

160,  835 

20    00 

1,  939,  527 

164,  506 

20 

00 

1,  919,  983 

167,  977 

20 

00 

1,899,853 

171,  241 

21    00 

2,  055,  460 

177,  280 

21    00 

2,035,605 

181,  324 

21 

00 

2,014,930 

185, 147 

21 

00 

1,993,740 

188,  741 

22    00 

2,152,302 

194,  518 

22    00 

2, 131,  338 

198,  953 

22 

00 

2, 109,  725 

203, 143 

22 

00 

2,  087,  468 

307,  086 

23    00 

2,  248,  998 

212,550 

33    00 

2,  227,  020 

217,  392 

23 

00 

3,  204,  359 

231,  966 

23 

00 

3, 181,  027 

336,  370 

24    00 

2,  345,  544 

231,  374 

24    00 

2,  322,  539 

236,  640 

24 

00 

2,  298,  825 

241,  616 

24 

00 

3,  274, 411 

246,  295 

25     00 

2,441,932 

260,  988 

25    00 

2,417,893 

256,  695 

25 

00 

2,  393, 116 

263,  089 

25 

00 

2,  367,  610 

267, 159 

26    00 

2,  538, 156 

271,  391 

26    00 

2,  513,  074 

277,  568 

26 

00 

2,  487,  224 

383,  383 

26 

00 

2, 460,  618 

288,  860 

27    00 

2,  634,  210 

292,  582 

27     00 

2,  608,  075 

299.  224 

27 

00 

2,581,144 

305,  498 

27 

00 

2,  653,  427 

311,  396 

28    m 

2,  730,  087 

314,  559 

28    00 

2,  702,  890 

321,  694 

28 

00 

3,  674,  867 

328,  432 

28 

00 

3,  646,  029 

334,  765 

29    00 

2,  825,  779 

337,  321 

29    00 

2,  797,  511 

344,  964 

29 

00 

2, 768,  385 

353, 183 

29 

00 

2,738,418 

358,966 

30    00 

2,  921,  284 

360,  866 

30    00 

2,891,931 

369,  036 

30 

00 

2,  861,  694 

376,  749 

30 

00 

2,830,585 

383,  997 

170 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XIX. — For  projections  of  maps  of  large  areas — Continued. 
[Extracteil  from  Appendix  No.  6,  U.  S.  Coast  and  Geodetic  Survey  Report  for  1884.] 


COORDINATES  OF  CURVATURE. 


N-ATXJEAL  SCALE.— VALUES  OE  X  AND  Y  IN  "METEES. 

Latitude  32°. 

Latitude  33°. 

Latitude  34°. 

Latitude  35°. 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

1    00 

94, 494 

437 

1 

00 

93,  454 

444 

1 

00 

92,  385 

451 

1 

00 

91,  289 

457 

2    00 

188,  980 

1,748 

2 

00 

186,  899 

1,777 

2 

00 

184,  762 

1,803 

2 

00 

182,  568 

1,  828 

3    00 

283,  449 

3,933 

3 

00 

280,  328 

3,  997 

3 

00 

277, 121 

4,057 

3 

00 

273,  830 

4,112 

4.    00 

377,  894 

6,991 

4 

00 

373,  731 

7,106 

4 

00 

369,  454 

7,212 

4 

00 

365,  064 

7,310 

5    00 

472,  307 

10,  922 

5 

00 

467, 100 

11, 102 

5 

00 

461,751 

11,268 

5 

00 

456,261 

11,  421 

6    00 

566.  680 

15,  727 

6 

00 

560,  428 

15,  986 

6 

00 

554,  004 

16,  225 

6 

00 

547,  412 

16,445 

7    00 

661,  004 

21, 404 

7 

00 

653,704 

21,757 

7 

00 

646,  205 

22,  082 

7 

00 

638,  509 

22,  381 

8    00 

755,  272 

27,954 

8 

00 

746,  922 

28,  414 

8 

00 

738,  344 

28,  839 

8 

00 

729,  542 

29,229 

9     00 

849,475 

35,  375 

9 

00 

840,  072 

35,  957 

9 

00 

830, 413 

36,  494 

9 

00 

820,  501 

36,  987 

10    00 

943,  605 

43,  667 

10 

00 

1,933,146 

44,385 

10 

00 

922,403 

45,  048 

10 

00 

911,  379 

45,  656 

11    00- 

1,  037, 655 

52,  829 

U 

00 

1,  026, 136 

53,  697 

11 

00 

1,  014,  305 

54,  499 

11 

00 

1,  002, 165 

55,  234 

.12    00 

1,131,616 

62,  861 

12 

00 

1.119,033 

63, 893 

12 

00 

1, 106, 110 

64,  846 

12 

00 

1,  092,  850 

65,  721 

13    00 

1,  225,  480 

73,  761 

13 

00 

1,  211.  829 

74,  971 

13 

00 

1, 197,  809 

76,  089 

13 

00 

1,  183,  426 

77, 115 

14    00 

1,319,239 

85,  529 

14 

00 

1,  304,  515 

86,  931 

14 

00 

1,  289,  395 

88,227 

14 

00 

1,273,834 

89,  415 

15    00 

1,412,885 

98, 164 

15 

00 

1,397,083 

99,  771 

15 

00 

1,  380,  858 

101,258 

15 

00 

1,  364, 214 

102, 619 

16    00 

1,506,411 

111,  664 

:  16 

00 

1,489,526 

113,491 

16 

00 

1, 472. 190 

115, 180 

16 

00 

1,454,407 

116,  728 

17    00 

1,  599,  808 

126,  029 

1  17 

00 

1,  581,  834 

128,  089 

17 

00 

1,  563,  381 

129,  993 

17 

00 

1,  544,  454 

131, 738 

18    00 

1,693,067 

141,  256 

18 

00 

1,  673,  998 

143,  564 

18 

00 

1,  654,  423 

145,  696 

18 

00 

1,634,347 

147,  650 

19    00 

1,  786, 182 

157,  346 

19 

00 

1,766,011 

159,  914 

19 

00 

1,745,308 

162,  287 

19 

00 

1,724,076 

164,460 

20    00 

1,  879, 144 

174,  296 

20 

00 

1,857,866 

177, 138 

20 

00 

1,  836,  026 

179,703 

20 

00 

1,  813,  632 

182, 168 

21    00 

1,971,946 

192, 105 

21 

00 

1,949,553 

195,  234 

21 

00 

1,  926,  569 

198, 124 

21 

00 

1,903,006 

200,  772 

22    00 

2,  064,  579 

210,  772 

22 

00 

2,041,062 

214,  201 

!  22 

00 

2,  016,  929 

217,  368 

22 

00 

1,  992, 190 

220,  268 

23     00 

2, 157.  035 

230,295- 

23 

00 

2, 132,  387  1  234,  037 

23 

00 

2, 107,  097 

237, 493 

23 

00 

2,  081, 174 

240,  657 

24     00 

2,  249,  305 

250.  672 

24 

00 

2,223,521     254,740 

24 

00 

2, 197,  065 

258, 497 

24 

00 

2, 169,  949 

261,  936 

25    00 

2,  341,  385 

271,901 

25 

00 

2,  314,  453 

276,  309 

!25 

00 

2, 286  823 

280,  378 

25 

00 

2,  258,  507 

284, 102 

26    00 

2,433,264 

293,  981 

26 

00 

2,405,175 

298,  741 

26 

00 

2,  376,  363 

303, 134 

26 

00 

2.  346,  838 

307, 154 

27    00 

2,524,935 

316,  910 

00 

2.  495,  080 

322,  034 

27 

00 

2, 465,  677 

326,  763 

27 

00 

2,434,934 

331,  089 

28    00 

2,  616,  390 

340,  686 

28 

00 

2,  585,  961 

346. 187 

28 

00 

2,  554,  756 

351,  262 

28 

00 

2,  522,  787 

355,  905 

29    00 

2,  707,  621 

.  365,  307 

29 

00 

2,  676,  007 

371, 197 

29 

00 

2,  643,  591 

376,  629 

29 

00 

2,610,386 

381,  598 

30    00 

2,  798,  621 

390,  770 

30 

00 

2,  765,  812 

397,061 

30 

00 

2,  732, 175 

402,  863 

30 

00 

2,  697,  724 

408,  168 

PEOJECTION  TABLES. 


171 


Table  XIX. — For  projections  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6,  TJ.  S.  Coast  and  Geodetic  Survey  Eeport  for  1884.] 

COORDINATES  OP  CUEVATUEE. 


NATUKAL  SCALE 

.—VALUES  OF  X  AND  T  METEPS. 

Latitude  36 

". 

Latitude  37°. 

Latitude  38=. 

Latitude  39°.  ■ 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

1    00 

90, 164 

462 

1 

00 

89,012 

467 

1 

00 

87,  833 

472 

° 
1 

00 

86,  627 

476 

2    00 

180,  319 

1,850 

2 

00 

178,  015 

1,870 

2 

00 

175,  656 

1,888 

2 

00 

173,  243 

1,903 

3    00 

270,455 

4,162 

3 

00 

266,  997 

4,207 

3 

00 

263, 458 

4,247 

3 

00 

259,  839 

4,281 

4,    OO 

360, 562 

7,399 

4 

00 

355,  951 

7,479 

4 

00 

351,  230 

7,549 

4 

00 

346,  403 

7,611 

5    00 

450,  631 

11,  560 

5 

00 

444,865 

11,685 

5 

00 

438,  962 

11,  795 

5 

00 

432,  925 

11,  891 

6    00 

540,  653 

16,  645 

6 

00 

533,  730 

16,824 

6 

00 

526,  643 

16,  983 

6 

00 

519,  396 

-  17, 121 

7    00 

630,  618 

22,  652 

7 

00 

622,  536 

22,  896 

7 

00 

614,  263 

23, 112 

7 

00 

605,  803 

23,  300 

8     00 

720,517 

29,583 

8 

00 

711,  273 

29,  901 

8 

00 

701,  812 

30, 183 

8 

00 

692. 138 

30,428 

9    00 

810,340 

37,435 

9 

00 

799,932 

37,838 

9 

00 

789,  280 

38, 195 

9 

00 

778,  388 

38,  504 

10    00 

900.  078 

46,  209 

10 

00 

888,  503 

46,  706 

10 

00 

876,  657 

47, 145 

10 

00 

864,545 

47,  527 

11     00 

989,  720 

55,903 

11 

00 

976,  975 

56,  503 

11 

00 

963,  933 

57,  034 

11 

00 

950,  598 

57,  496 

13    00 

1, 079,  259 

66,  515 

12 

00 

1,  065,  34Q 

67,  229 

12 

00 

1,  051,  098 

67,  860 

12 

00 

1,  036,  536 

68,  409 

13     00 

1, 168,  684 

78,  046 

13 

00 

1, 153,  587 

78, 882 

13 

00 

1,  138. 141 

79,  622 

13 

00 

1, 122,  349 

80,  266 

U    00 

1,  257,  987 

90,  494 

14 

00 

1,  241,  707 

91, 462 

14 

00 

1.225,053 

92,319 

14 

00 

1,  208,  027 

93,  064 

15    00 

1,  347, 156 

103,  856 

15 

00 

1,  329,  690 

104,  967 

15 

00 

1,  311,  823 

105,  949 

15 

00 

1,  293,  559 

106,802  1 

16    00 

1,  436, 184 

118, 133 

16 

00 

1, 417,  526 

119,  395 

16 

00 

1,  398,  441 

120,  511 

16 

00 

1.  378,  934 

121,  479  1 

17     00 

1,  525,  061 

133,  323 

17 

00 

1,  505,  206 

134,745 

17 

00 

1,484,899 

136,  002 

17 

00 

1,  464, 144 

137,093  ' 

18    00 

1,  613,  777 

149,  423 

18 

00 

1,592,721 

151,  015 

18 

00 

1,  571,  183 

152,  421 

18 

00 

1,  549, 177 

153,642 

19    00 

1,  702,  324 

166, 433 

19 

00 

1, 680,  059 

168,  203 

19 

00 

1,  657,  289 

169,  767 

19 

00 

1,  634,  023 

171,124 

20    00 

1,790,691 

184,  3.50 

20 

00 

1,  767,  211 

186,  307 

20 

00 

1,  743,  202 

188,  037 

20 

00 

1,  718,  671 

189,537 

21    00 

1,  878,  870 

203, 173 

21' 

00 

1,  854, 169 

205,  326 

21 

00 

1,  828.  914 

207,  229 

21 

00 

1,  803, 113 

208,878 

22    00 

1,  966,  851 

222,  899 

22 

00 

1,  940,  922 

225,  258 

22 

00 

1,  914,  415 

227,  341 

22 

00 

1,  887,  337 

229, 146 

23    00 

2,  054,  625 

243,527 

23 

00 

2,027,462 

246,  099 

23 

00 

1,  999,  694 

248,370 

23 

00 

1,  971,  333 

250,337 

24    00 

2, 142, 183 

265,  055 

24 

00 

2, 113,  777 

267,  849 

24 

00 

2,  084,  743 

270, 315 

24 

00 

2,  055,  091 

272,  450 

25    00 

2,  229,  516 

287,  479 

25 

00 

2, 199,  860 

290,  503 

25 

00 

2,169,551 

293, 172 

25 

00 

2, 138,  602 

295, 481 

26    00 

2,  316,  613 

310,  798 

26 

00 

2,  285,  699 

314,  061 

26 

00 

2,  254, 109 

316,  939 

26 

00 

2,221,854 

319.  429 

27    00 

2.403,467 

335,  009 

27 

00 

2,  371,  287 

338,  519 

27 

00 

2,  338,  406 

341, 613 

27 

00 

2, 304,  838 

344,  289 

28    00 

2,490,068 

360.  Ill 

26 

00 

2,  456.  6l2 

363,  874 

28 

00 

2,  422,  433 

367, 192 

28 

00 

2,  387,  545 

370,  059 

29    00 

2,  576, 407 

386,  099 

29 

00 

2,  541,  667 

390, 125 

29 

00 

2,  506, 181 

393,  672 

29 

00 

2,  469,  963 

396,  736 

30    00 

2,  662, 475 

412,  971 

30 

00 

2,  626,  441 

417,  267 

30 

00 

2,  589,  639 

421,  050 

30 

00 

2,  552,  084 

424,  317 

172 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XIX. — For  projections  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6.  V.  S.  Coast  and  Geodetic  Survey  Report  for  1884.] 

COOEDINATES  OP  CnEVATUEE. 


NATURAL  SCALE.- 

VALUES  OF  X  AND  T  IN  METERS, 

Latitude  40 

". 

Latitude  41°. 

Latitude  42°. 

Latitude  43 

"■ 

Longi- 
tude. 

X 

T 

Longi- 
tude. 

s 

T 

Longi- 
tude. 

X 

T 

Longi- 
tude. 

X 

Y 

1    00 

85,  394 

479 

1 

00 

84, 136 

482 

1 

00 

82,  851 

484 

1 

00 

81, 541 

485 

2    00 

170,  778 

1,916 

2 

00 

168,  260 

1,927 

2 

00 

165,  691 

1,935 

2 

00 

163,  071 

1,941 

3     00 

256, 140 

4,311 

3 

00 

252,  363 

4,335 

3 

00 

248,  508 

4,354 

3 

00 

244,  578 

4,367 

4    00 

,     341,470 

7,663 

4 

00 

336, 432 

7,706 

4 

00 

331,  292 

7,739 

4 

00 

326,  050 

7,763 

5    00 

426,  757 

11,  972 

5 

00 

420,457 

12,  039 

5 

00 

414,  030 

12,  092 

5 

00 

407,  476 

12, 129 

6    00 

511, 990 

17,  238 

6 

00 

504, 428 

17,  335 

6 

00 

496,  712 

17,410 

6 

00 

488,  844 

17,  464 

7     00 

597, 158 

23,400 

7 

00 

588,  332 

23,  591 

7 

00 

679,  325 

23,  693 

7 

00 

570, 143 

23,766 

8     00 

682,  252 

30,  637 

8 

00 

672, 159 

30, 807 

8 

00 

661,  861 

30,941 

8 

00 

651,  361 

31,  036 

9     00 

767,260 

38,  768 

9 

00 

755,897 

38,  983 

9 

00 

744,  305 

39, 152 

9 

00 

732,  486 

39,  272 

10     00 

852, 171 

47,  852 

10 

00 

-    839,537 

48, 118 

10 

00 

826,  648 

48,  325 

10 

00 

813,  508 

48.  474 

11     00 

936,  975 

57,  888 

11 

00 

923,  067 

58,  209 

11 

00 

908.  879 

58,459 

11 

00 

894,  415 

58,  639 

12     00 

1,021,661 

68,  875 

12 

00 

1,006,475 

69,  256 

12 

00 

.990,985 

69,  553 

12 

00 

975, 195 

69,  766 

13     00 

1, 106,  218 

80,611 

13 

00 

1,  089,  752 

81,  258 

13 

00 

1,  072,  956 

81,  605 

13 

00 

1,  055,  837 

81,854 

14    00 

1,190,636 

93,  695 

14 

00 

1, 172,  886 

94, 212 

14 

00 

1, 154, 781 

94,  614 

14 

00 

1, 136,  329 

94,901 

15    00 

1, 274, 904 

107,  525 

15 

00 

1,255,866 

108, 117 

15 

00 

1,236,449 

108,  577 

15 

00 

1,  216,  661 

108,  905 

16    00 

1,  359.  012 

122,  300 

16 

00 

1,338,681 

122,  971 

16 

00 

1,  317,  948 

123,493 

16 

00 

1,  296,  820 

123,  864 

17    00 

1,  442,  949 

138, 017 

17 

00 

1, 421,  321 

138,  773 

17 

00 

1,  899, 267 

139,  360 

17 

00 

1,  376,  795 

139,  777 

18    00 

1,  526,  704 

154,  675 

18 

00 

1,  503, 775 

155,  520 

18 

00 

1, 480,  395 

156, 175 

18 

00 

1,  456,  575 

156,  640 

19    00 

1,  610,  267 

172,  272 

19 

00 

1,  586,  031 

173,  210 

19 

00 

1, 561,  321 

173,  937 

19 

00 

1,  536, 148 

174,451 

20    00 

1,  693.  623 

190,  805 

20 

00 

1,  608,  079 

191.  841 

20 

00 

1,  642,  035 

192,  642 

20 

00 

1,  615,  505 

193,  209 

21     00 

1,  776,  775 

210,  272 

21 

00 

1,  749.  909 

211,  409 

21 

00 

1,  722,  524 

212,  289 

21 

00 

1,  694,  632 

212,  909 

22     00 

1.  8S0,  698 

230,  671 

2'^ 

00 

1,831,509 

231,  914 

22 

00 

1,802,779 

232.  874 

22 

00 

1,  773,  519 

233,  551 

23    00 

1,  942,  387 

251,998 

23 

00 

1,  912,  869 

253,  352 

23 

00 

1,  882,  788 

254,  396 

23 

00 

1,  852, 135 

255,129 

24    00 

2,  024, 833 

274,  252 

24 

00 

1,  993,  978 

275,  719 

24 

00 

1,  962,  540 

276,  850 

24 

00 

1,  930,  528 

277,  642 

23    00 

2, 107,  023 

297.  430 

25 

00 

2,  074,  826 

299,  014 

25 

00 

2,  042,  024 

300,  234 

25 

00 

2,  008,  628 

301,  087 

26    00 

2. 188,  948 

321,  528 

26 

00 

2,1.55,402 

323,  233 

26 

00 

2,121  230 

324,  544 

26 

00 

2,  086,  443 

325,  459 

27     00 

2,  270,  597 

346,  543 

27 

00 

2,  235,  695 

348,  374 

27 

00 

2,  200, 146 

349,  778 

27 

00 

2, 163,  963 

350,  750 

28    00 

2,  351,  961 

372,  473 

28 

00 

2,  315,  695 

374,  432 

28 

00 

2,  278,  762 

375,  932 

28 

00 

2,  241, 176 

376,  974 

29     00 

2,433,029 

399,  314 

29 

00 

2,  395,  392 

401, 404 

29 

00 

2,  357,  067 

403,  002 

29 

00 

2,  318,  071 

404, 109 

30    00 

2,513,790 

427,  063 

30 

00 

2,474,774 

429,  287 

30 

00 

2,  435,  052 

430,  985 

30 

00 

2,  394,  639 

432, 157 

PEOJBCTION  TABLES. 


173 


Table  XIX.- — For  projections  of  maps  of  large  areas — Continued. 
[Extracted  from  Appendix  No.  6,  U.  S.  Coast  and  Geodetic  Survey  Eeport  for  1884.] 


COORDINATES  OP  CURVATURE. 


NATURAL  SCALE.- 

-VALUES  OF  X  AND  Y  TS  METERS. 

Latitude  44°. 

Latitude  45°. 

L.atitude  46°. 

Latitude  47°.              1 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Longi- 
tude. 

X 

Y 

Lo 
tu 

1 

agi- 
de. 

X 

Y 

1    00 

80,  206 

486 

1 

00 

78,  847 

486 

1 

00 

77,464 

486 

00 

76,  056 

485 

2    00 

160,  401 

1,945 

2 

00 

157,  682 

1,946 

2 

00 

154,915 

1,945 

2 

00 

152, 100 

1,942 

3    00 

240,  572 

4,375 

3 

00 

236,  493 

4,378 

3 

00 

232, 342 

4,376 

3 

00 

228, 119 

4,368 

4    00 

320,  708 

7,778 

4 

00 

315,  269 

7,783 

4 

00 

309,  732 

7,779 

4 

00 

304, 101 

7,765 

5    00 

400,  797 

12, 152 

5 

00 

393,  996 

12, 160 

5 

00 

387,  074 

12, 153 

5 

00 

380,  034 

13,131 

6    00 

480,  82' 

17,496 

6 

00 

472,  663 

17,  508 

6 

00 

464,  354 

17,  498 

6 

00 

455,  904 

17, 467 

7    00 

560, 786 

23,  811 

7 

00 

551,  258 

23,  826 

7 

00 

541,  562 

23,813 

7 

00 

531,  700 

23,  770 

8    00 

640,  062 

31,094 

8 

00 

629,  769 

31, 114 

8 

00 

618,684 

31,  096 

8 

00 

607, 410 

31,  040 

9    00 

720,445 

39,  345 

9 

00 

708, 184 

39,  370 

9 

00 

695,  708 

39,  347 

9 

00 

683,  020 

39,  276 

10    00 

800, 122 

48,  563 

10 

00 

786,492 

48,  594 

10 

00 

772,  623 

48,565 

10 

00 

758,  520 

48,477 

11    00 

879,  681 

58,  746 

11 

00 

864, 679 

58,  782 

11 

00 

849, 416 

58,  747 

11 

00 

833,  895 

58,  640 

12    00 

959, 110 

69,  893 

12 

00 

942, 735 

69,  936 

12 

00 

926,  075 

69,  893 

12 

00 

909, 135 

69,  765 

13    OO 

1,  038,  399 

82,  002 

13 

00 

1,  020,  647 

82,  051 

13 

00 

1,002,588 

82,  000 

13 

00 

984,  227 

81,  849 

14    00 

1, 117,  535 

95,  072 

14 

00 

1,098,404 

95, 127 

14 

00 

1,  078,  943 

95,  067 

14 

00 

1,059,158 

94,  890 

15    00 

1, 196,  507 

109,100 

15 

00 

1, 175,  994 

109, 162 

15 

00 

1, 155, 128 

109,  091 

15 

00 

1, 133,  917 

108,  887 

16    00 

1,  275,  303 

124,  084 

16 

00 

1,  253,  404 

124, 153 

16 

00 

1,  231, 131 

124, 071 

16 

00 

1,  208, 491 

123,837 

17    00 

1,  353,  911 

140,  023 

17 

00 

1,330,634 

140,  099 

17 

00 

1,306,940 

140, 003 

17 

00 

1,  282,  868 

139, 738 

18    00 

1,  432,  320 

156,913 

18 

00 

1,407,640 

156,  996 

18 

00 

1,  382,  543 

156,  887 

18 

00 

1,  357,  036 

156,  587 

19    00 

1, 510,  519 

174,  753 

19 

00 

1,  434, 443 

174,  842 

19 

00 

1,457,928 

174,  718 

19 

00 

1, 430,  984 

174,  381 

20    00 

1,  588,  496 

193,  540 

20 

00 

1,  561,  019 

193,  635 

20 

00 

1,533,083 

193,  494 

20 

00 

1,  504,  697 

193, 118 

21    00 

1,  666,  240 

213,270 

21 

00 

1,  637,  358 

213,  .371 

21 

00 

1,  607,  997 

213,  212 

21 

00 

1,  578, 166 

212,  793 

22    00 

1,  743,  738 

233,  942 

22 

00 

1,  713,  447 

234,  048 

22 

00 

1.682,657 

233,  869 

22 

00 

1,  651,  377 

233,  405 

23    00 

1,820,980 

255,  552 

23 

00 

1,  789,  276 

255,  663 

23 

00 

1,  757,  052 

255, 462 

23 

00 

1,  724,  320 

254,  950 

24    00 

1,  897,  955 

278,  096 

24 

00 

1,  864,  831 

278,  211 

24 

00 

1,  831, 170 

277,  987 

24 

00 

1,  796,  982 

277,  425 

25    00 

1,  974,  650 

301,  572 

25 

00 

1,  940, 103 

301,  690 

25 

•00 

1,  904,  999 

301,  441 

25 

00 

1,  869,  3.51 

300,  824 

26    00 

2, 051,  055 

325, 977 

26 

00 

2,  015,  079 

326,  097 

26 

00 

1,  978,  528 

325,  820 

26 

00 

1,941,415 

325, 146 

27    00 

2, 127, 159 

351,  306 

27 

00 

2,  089, 749 

351, 427 

27 

00 

2,  051,  745 

351, 120 

27 

OO 

2,  013, 163 

350,  386 

28    00 

2,  202,  950 

377,  555 

28 

00 

2, 164, 100 

377,  676 

28 

00 

2, 124,  639 

377, 337 

38 

00 

2,  084,  583 

376,  539 

29    00 

2,  278,  417 

404,  722 

29 

00 

2,  238, 121 

404,  841 

29 

00 

2, 197, 197 

404,  468 

29 

00 

2, 155,  663 

403,  602 

30     00 

2,  353,  550 

432,  801 

30 

00 

2,  311,  802 

432,  918 

30 

00 

2,  269,  410 

432,  507 

30 

00 

2,226,392 

431,  569 

174 


A  MAXUAL  OF  TOrOGEAPHIC  METHODS. 


Table  XIX. — For  projections  of  maps  of  large  areas-^Continned. 

[Extracted  from  Appendix  No.  6,  U.  S.  Coast  and  Geodetic  Survey  Keport  for  1884.] 

COORDINATES  OF  CnEVATUEK. 


NATURAL  SCALE.- VALUES  OF  X  AND  T  IN  METERS. 

Latitude  48°. 

Latitude  49°. 

Latitude  50°. 

Longi- 
tude. 

x 

y 

Longi- 
tude. 

X 

Y   ' 

Longi- 
tude. 

X 

Y 

1    00 

74,  626 

484 

1 

00 

73, 172 

482 

1 

00 

71,  696 

479 

2    00 

149.  239 

1,936 

2 

00 

146,  331 

1,928 

2 

00 

143,  379 

1,917 

3    00 

223,  827 

4,355 

3 

00 

219,  465 

4,337 

3 

00 

215,  037 

4.313 

4    00 

298, 377 

7.742 

4 

00 

292,561 

7,709 

4 

00 

286,  656 

7,667 

5     00 

372,877 

12,  095 

5 

00 

365,  606 

12,044 

5 

00 

358,  224 

11,  978 

6    00 

447, 314 

17,414 

6 

00 

438,  588 

17.  340 

6 

00 

429,  727 

17,246 

7    GO 

521,  677 

23,  698 

7 

00 

511,493 

23,  598 

7 

00 

501, 154 

23,  469 

8     00 

595,  951 

30,  946 

8 

00 

584,  310 

30,  815 

8 

00 

572,  492 

30,646 

9    00 

670, 125 

39, 157 

9 

00 

657,  026 

38,  991 

9 

00 

643,  727 

38,  777 

10    00 

744, 186 

48,  329 

10 

00 

729,  627 

48, 123 

10 

00 

714,  847 

47,  859 

11    OD 

.     818, 123 

58,  461 

11 

00 

802, 102 

58,  212 

11 

00 

785,  839 

57,  891 

12    00 

891,  921 

69,  552 

12 

00 

874, 438 

69,  254 

12 

00 

856,  691 

68,872 

13    00 

965, 570 

81.  598 

13 

00 

946,  622 

81,  248 

13 

00 

927,  389 

80, 798 

14    00 

1,039,056 

94,  598 

14 

00 

1,018,642 

94, 191 

14 

00 

997,  922 

93, 669 

15    00 

1,112,367 

108,  551 

15 

00 

1,  090,  485 

108,  082 

15 

00 

1,  068,  277  ■ 

107,  482 

16    00 

1, 185, 491 

123, 453 

16 

00 

1, 162, 138 

122,  918 

16 

00 

1, 138,  440 

122,  234 

17    00 

1,258,416 

139,  302 

17 

00 

1,  233,  591 

138,  697 

17 

00 

1,208,400 

137,  923 

18    00 

1,  331, 129 

156,  096 

18 

00 

1,  304, 829 

155,  416 

18 

00 

1,  278, 144 

154,  546 

19    00 

1,  403,  618 

173, 832 

19 

00 

1,375,840 

173,  071 

19 

00 

1,  347,  660 

172,  099 

20    00 

1,  475,  871 

192,  506 

20 

00 

1,  446,  613 

191,  660 

20 

00 

1,  416,  934 

190,  581 

21     00 

1,  547,  876 

212, 116 

21 

00 

1,  517, 135 

211, 180 

21 

00 

1,  485,  956 

209,  987 

22    00 

1,  619,  620 

232, 658 

22 

00 

1,  587,  394 

231,  627 

22 

00 

1, 554,  711 

230,  314 

23    00 

1,  691,  091 

254, 128 

23 

00 

1,  657,  378 

252,998 

23 

00 

1,  623, 189 

251,  559 

24    00 

1,  762,  279 

276,  524 

24 

00 

1,727,073 

275,  288 

24 

00 

1,  691,  377 

273, 717 

25     00 

1,833,170 

299,  842 

25 

00 

1,  796,  470 

298,  495 

25 

00 

1,759,262 

296,  785 

26    00 

1,903,752 

324,  077 

26 

00 

1,  865,  554 

322,  614 

26 

00 

1,  826,  833 

320,  758 

27    00 

1,  974.  015 

349,  225 

27 

00 

1,  934,  315 

347,  640 

27 

00 

1,894,077 

345,  633 

28    00 

2,  043,  945 

375,  283 

28 

00 

2,  002,  740 

373, 570 

28 

00 

1,  960,  983 

371,404 

29    00 

2, 113,  531 

402,  245 

29 

00 

2,  070,  817 

400,  399 

29 

00 

2, 027,  538 

398,  068 

30     00 

2, 182,  762 

430, 107 

30 

00 

2, 138,  536 

428, 123 

30 

00 

2,  093,  731 

425,  619 

PROJECTION  TABLES. 


175 


Table  XX. — Cooi-dinates  for  projection  of  maps.     Scale  ^j-ooos- 
[Prepared  by  R.  S.  Woodward.] 


a  " 


Coordinates  of  developed  parallel  for — 


Inches. 

'"4.36i' 
8. 723 
13.  083 

17. 444 
4.362 
8. 723 

13. 085 


4.362 
■  8.  724 
13. 087 


4.363 
8.726 
13.  088 


13. 091 
17.454 
4.364 
8.728 
13.  092 


4.  365 
8.730 
13. 095 


4.367 
8.734 
13.101 


4.368 
8.735 
13. 103 


3.750 
3.740 
3.730 


3.679 
3.669 


3.583 
3.572 
3.561 


.004 
.004 


longitude.  45'  longitude.  1°  longitude, 


.004 
.004 
.004 

.004 
.004 


7.949 
7.933 
7.916 
7.900 


7.798 
7.780 
7.763 


7.727 
7.709 
7.691 

7.673 
7.654 


7.578 
7.559 
7.540 

7.520 
7.500 


7.420 
7.400 
7.379 


7.253 
7.231 
7.210 


7.166 
7.144 
7.122 


.017 
.017 
.017 


.018 
.018 


.018 
.018 
.018 


Inches. 
11.923 
11.899 
11.  874 
11.850 

11.  825 


11.  697 
11.671 
11.644 


11.  591 
11.  563 
11. 536 


11.481 
11. 453 
11.425 


11. 367 
11. 338 
11.  309 


11.250 
11. 221 
11. 191 


11. 130 
11. 100 
11.069 


11. 007 
10.  975 
10. 943 


10. 879 
10. 847 
10. 815 


10. 749 
10.  716 
10.  683 


.040 
.040 


Inches. 
15.  898 
15.  865 
15.  832 
15.  800 

15.707 
15. 733 
15. 699 
15.  665 


15. 596 
15.  561 
15.  526 


15.  454 
15.418 
15. 382 


15. 156 
15. 118 
15. 079 


15.  001 
14.  961 
14. 921 


14. 840 
14. 799 
14.  758 


14.  676 
14. 633 
14. 591 


14.  506 
14. 463 
14.  420 


14. 332 
14.  288 
14.244 


4.369 
8.738 
13. 108 


3.539 
3.527 
3.516 


7.077 
7.054 
7.032 


6.986 
6.963 
6.939 


10.  616 
10.  582 
10. 547 


10.  479 
10.444 
10.  409 


.041 
.041 
.041 

.041 


14. 154 
14. 109 
14.  063 


13.  972 
13.  925 
13.  879 


176 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XS. — Coordiiiates  for  projection  of  maps.    Scale  ■ 
[Prepared  by  E.  S.  Woodward.] 


ft 

Coordinates  of  developed  parallel  for— 

1 

15'  longitude. 

30'  longitude. 

45'  longitude. 

lo  longitude. 

X 

y 

s 

y 

- 

y 

X 

y 

38    00 

15 

30 
45 

Inches. 
17.  477 

Inches. 
3.458 

3.446 
3.434 
3.422 

Inches. 
.005 

.005 
.005 
.005 

iTiches. 
6.916 

6.892 
6.809 
6.845 

Inches. 
.019 

.019 
.019 
.019 

Inches. 
10. 374 

10. 339 
10. 303 
10.267 

Inches. 
.042 

.042 
.042 
.042 

Inches. 
13. 832 

13.785 
13. 737 
13. 690 

Inches. 
.074 

.074 
.075 
.075 

4.370 
8.740 
13. 110 

39    00 

15 
30 
45 

17.  480 

3.411 

3.398 
3.386 
3.374 

.005 

.005 
.005 
.005 

6.821 

6.797 
6.773 
6.748 

.019 

.019 
.019 
.019 

10.232 

10. 195 
10. 159 
10. 123 

.042 

.042 
.042 
.042 

13.642 

13.  594 
13.545 
13.497 

.075 

.075 
.075 
.075 

4.371 
8.741 
13. 112 

40    00 

15 
30 

45 

17.483 

3.362 

3.350 
3.337 
3. 325 

.005 

.005 
.005 
.005 

6.724 

6.699 

6.675 
6.650 

.019 

.019 
.019 
.019 

10.  086 

10.049 
10.  012 
9.975 

.042 

.042 
.043 
.043 

13.448 

13.399 
13.  349 
13.  300 

.075 

.075 
.076 
.076 

4.371 
8.743 
13. 114 

41    00 

17.  486 

3.312 

.005 

6.625 

.019 

9.937 

.043 

13.  250 

.076 

15 
30 
45 

4.372 
8.744 
13. 117 

3.300 
3.287 
3.275 

.005 
.005 
.005 

6.600 
6.575 
6.549 

.019 
.019 
.019 

9.900 
9.862 
9.824 

.043 
.043 
.043 

13.  200 
13.149 
13.  098_ 

.076 
.076 
.076 

42    00 

15 
30 

45 

17.489 

3.262 

3.249 
3.236 
3.223 

.005 

.005 
.005 
.005 

6.524 

6.498 
6.472 
6.447 

.019 

.019 
.019 
.019 

9.786 

9.747 
9.709 
9.670 

.043 

.043 
.043 
.043 

13.048 

12.  996 
12.  945 
12.  893 

.076 

.076 
.076 
.076 

4.373 
8.746 
13. 119 

43    00 

15 
30 
45 

17.  492 

3.210 

3.197 
3.184 
3.170 

.005 

.005 
.005 
.005 

6.421 

6.394 
6.368 
6.342 

.019 

.019 
.019 
.019 

9.631 

9.592 
9.552 
9.513 

.043 

.043 
.043 
.043 

12.842 

12.  789 
12.  736 
12.684 

.076 

.076 
.076 
.076 

4.374 
8.747 
13. 121 

44    00 

15 

30 

45 

17.495 

3.158 

3.144 
3.131 
3.118 

.005 

.005 
.005 
.005 

6.316 

6.289 
6.262 
6.235 

.019 

.019 
.019 
.019 

9.473 

9.433 
9.393 
9.353 

.043 

.043 
.043 
.043 

12.631 

12.578 
12.524 
12.471 

.077 

.077 
.077 
.077 

4.375 
8.749 
13.  124 

45    00 

15 
30 
45 

17.  498 

3.104 

3.091 
3.077 
3.063 

.005 

.005 
.005 
.005 

6.209 

6.181 
6.154 
6.127 

.019 

.019 
.019 
.019 

9.313 

9.272 
9.231 
9.190 

.043 

.043 
.043 
.043 

12.417 

12.363 
12.  308 
12.254 

.077 

.077 
.077 
.077 

4.375 
8.751 
13. 126 

46    00 

15 
30 
45 

17.  501 

3.050 

3.036 
3.022 
3.008 

.005 

.005 
.005 
.005 

6.100 

6.072 
6.  044 
6.017 

.019 

.019 
.019 
.019 

9.150 

9.108 
9.067 
9.025 

.043 

.04a 
.043 
.043 

12.200 

12.144 

12.  089 
12.  033 

.077 

.077 
.077 
.077 

4.376 
8.752 
13. 128 

47    00 

15 
30 
45 

17.  504 

2.994 

2.980 
2.966 
2.962 

.005 

.005 
.005 
.005 

5.989  * 

5.961 
5.933 
5.904 

.019 

.019 
.019 
.019 

8.983 

8.941 
8.899 
8.857 

.043 

.043 
.043 
.043 

11.  978 

11.922 
11.865 
11.809 

.076 

.076 
.076 
.076 

4.377 
8.754 
13. 131 

f48    00 

15 
30 

45 

17.508 

2.938 

2.924 
2.909 
2.895 

.005 

.005 
.005 
.005 

5.876- 

5.848 
5.819 
5.790 

.019 

.019 
.019 
.019 

8.814 

8.771 
8.728 
8.686 

.043 

.043 
.043 
.043 

11.  752 

11.695 
11.  638 
11.  581 

.076 

.076 
.076 
.076 

4.378 
8.755 
13. 133 

^49    00 

15 
30 
45 

17.  511 

2.881 

2.866 
2.852 
2.837 

.005 

.005 
.005 
.005 

5.762 

5.733 
5.704 
5.675 

.019 

.019 
.019 
.019 

8.643 

8.  599 
8.555 
8.512 

.043 

.043 
.043 
.042 

11.524 

11.465 
11.407 
11.349 

.076 

.076 
.076 
.076 

4.378 
8.757 
13. 135 

50    00 

17.514 

2.823 

.005 

5.646 

.019 

8.468 

.042 

11.291 

.076 

PEOJEOTION  TABLES. 


177 


Table  XXI. — Coordinates  for  projection  of  maps.    Scale  ti^outt- 
[Prepared  by  E.  S.  Woodward.] 


Abscissas  of  developed  parallel. 


25'  longi-   30'  long! 
tude.  tude. 


Ordinates  of  devel- 
oped parallel. 


Inches. 

"Km 

11.629 
17.444 
23. 259 
29.  074 


5.816 
11.  633 
17.  449 
23.  265 
29.  082 


5.817 
11.  634 
17.  451 
23.  268 
29.  086 


5.818 
11.  636 
17.  454 
23.  272 
29.  090 


11.  638 
17.  457 
23.  276 
29.  094 


11.  640 
17. 460 
23.  280 
29. 100 


5.821 
11.642 
17.  462 
23.  283 
29, 104 


6.822 
.1.  643 
17.  465 
23.  287 
29. 109 


5.823 
11.  645 
17.468 
23.  291 
29. 113 


2.642 
2.  639 
2.635 
2.631 

2,628 
2.  624 
2,620 
2.616 
2,613 
2,609 

2,605 
2.601 
2.597 
2.593 
2.589 
2.586 

2.582 
2,578 
2,574 
2,570 
2.566 
2.662 

2.558 
2.553 
2,549 
2.545 
2.541 
2.537 

2,533 
2,528 
2,524 
2,520 
2.515 
2.511 

2.507 
2.502 
2,498 


2.480 
2.476 
2.471 
2,467 
2,462 
2,458 

2.453 
2,448 
2,  444 
2,439 
2,434 


2,  425 
2,420 
2,415 
2.410 
2,406 
2,401 


Inches. 
5.299 
5,292 
5.285 
5,278 
6,270 
5.263 

5.256 
6.248 
6,240 
5.233 
5,225 
5.218 

5,210 
5.203 
5.195 
5.187 
5.179 
5.171 

5.163 
5.155 
5.147 
5.139 
5.131 
5.123 


5.065 
5,056 
5.048 
5.  039 
5.031 
5.022 


4.951 
4.942 
4,933 
4.924 
4.916 


4.821 
4.811 
4.802 


Inches. 
7.949 
7.938 
7.927 


7.883 
7.872 
7.861 


7.804 
7.792 
7.780 
7,768 
7,757 

7.745 
7,733 
7,721 
7,709 
7,697 
7.685 

7.673 
7.660 
7.648 
7.635 
7.622 
7.610 


7.559 
7.546 
7.533 

7,520 
7,507 
7.494 


7.441 
7.427 
7.413 
7,400 
7.386 
7.373 

7,359 
7,345 
7.331 
7,316 
7,302 


7.274 
7.260 
7,246 
7,231 
7.217 
7.203 


Inches. 
10. 699 
10,  584 
10,  670 
10,  555 
10.  540 
10,  526 

10,  511 
10,  496 
10.  481 
10,  466 
10,  451 
10,  436 

10,  421 
10,  405 
10,  390 
10.  37i 
10,  368 
10.  342 

10. 327 
10.  311 
10.  291 
10.  278 
10.  262 
10.  246 

10. 230 
10,  213 
10. 197 
10. 180 
10. 163 
10. 146 

10, 130 
10. 113 
10,  096 
10,  078 
10,  061 
10,  044 

10.  027 
10.  009 
9,992 
9,974 


9,774 
9.755 
9.736 
9.718 


9.661 
9.642 
9,622 


Indies. 
13. 249 
13,  231 
13,  212 
13, 194 
13, 176 
13. 157 

13, 139 
13, 120 
13, 101 
13,  082 
13.  063 
13. 045 

13.  026 
13.  006 
12,  987 
12. 967 
12.  947 
12. 928 

12,  909 
12,  889 
12,  868 
12,  848 
12,  828 
12.  808 

12,  788 
12,  767 
12.  746 
12,  725 
12,  704 
12.  683 

12.  662 
12,  641 
12,  620 
12.  598 
12, 577 
12.  556 

12.  534 
12.  512 
12. 490 
12,  467 
12,445 
12. 423 

12.  401 
12.  379 
12.  356 
12.  333 
12.  310 
12.  388 

12.  265 
12.  241 
12.  218 
12, 194 
12. 171 
12. 147 

12. 124 
12, 100 
12,  076 
12,  052 
12,  028 
12,  004 


Inches. 
15, 898 
15,  877 

15,  854 

16.  833 

15.  811 

16.  788 

15.  767 
15. 744 

16.  721 
15.  698 
15,  676 
15.  664 

15.  631 
15.  608 
15.  584 
15,  560 
15,  537 
15. 514 

15.  490 
■16.  466 
15,442 
15,  418 
15.  394 

15,  369 

16,  346 
16,  320 
15,  295 
15. 270 
15.  246 
15.  220 

15. 195 
16. 169 
15. 143 
15. 118 
15.  092 
15,  066 

15,  040 
15,  014 
14,  987 
14.  960 
14.  934 
14,  908 

14.  881 
14.  854 
14,  827 
14. 800 
14.  772 
14.  745 

14.  717 
14.  689 
14.  661 
14.  633 
14.  605 
14.  575 

14.  549 
14.  620 
14.  491 
14.  462 
14. 434 
14. 405 


0.001 
.004 
.008 


-12 


178  A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

Table  XXl.— Coordinates  of  projection  of  maps.    Scale  t^sWd — Continued. 
[Prepared  by  K.  S.  'Wooilw.ircl.] 


|1 
i3 

§  «  i'i 

'C  ta  a  S 

Abscissas  of  developed  parallel. 

Ordinates  of  devel- 
oped parallel. 

5'  longi- 
tude. 

0'  longi- 
tude. 

15'  longi- 
tude. 

20'  longi- 
tude. 

^5'  longi- 
tude. 

30'  longi- 
tude. 

35  00 
10 
20 
30 
40 
50 

36  00 
10 
20 
30 
40 
50 

37  "0 
10 
20 
30 
40 
50 

38  00 
10 
20 
30 
40 
50 

39  00 
10 
20 
30 
40 
50 

40  00 
10 
20 
30 
40 
50 

41  00 
10 
20 

1             30 
40 
50 

42  00 
10 
20 
30 

40 
50 

43  00 
10 
20 
30 
40 
50 

Inches. 

Inches, 
2.396 
2.391 
2.386 
2.3S1 
2.377 
2.372 

2.367 
2.362 
2.357 
2.351 
2.346 
2.  341 

2.336 
2.331 
2.326 
2.321 
2.316 
2.311 

2.305 
2.300 
2.295 
2.290 
2.284 
2.279 

2.274 
2.268 
2.263 
2.258 
2.252 
2.247 

2.241 
2.236 
2.230 
2.  225 
2.219 
2.214 

2.208 
2.203 
2.197 
2.192 
2.186 
2.180 

2.175 
2.169 
2.163 
2.157 
2. 152 
2.146 

2.140 
2.135 
2.129 
2.123 
2.117 
2.111 

I7iches. 
4.792 
4.782 
4.773 
4.763 
4.753 
4.743 

4.733 
4.723 
4.713 
4.703 
4.693 
4.683 

4.673 
4.  662 
4.652 
4.642 
4.631 
4.621 

4.611 
4.600 
4.590 
4.579 
4.568 
4.558 

4.548 
4.537 
4.526 
4.515 
4.504 
4.493 

4.483 
4.472 
4.461 
4.450 
4.439 
4.428 

4.417 
4.406 
4.394 
4.383 
4.372 
4.360 

4.349 
4.338 
4.326 
4.315 
4.303 
4.292 

4.281 
4.269 
4.257 
4.246 
4.234 
4.222 

Inches. 
7.188 
7.174 
7.159 
7.144 
7.130 
7.115 

7.099 
7.085 
7.070 
7.055 
7.  039 
7.024 

7.009 
6.994 
6.978 
6.963 
6.947 
6.932 

6.916 
6.900 
6.884 
6.869 
6.853 
6.837 

6.821 
6.805 
6.789 
6.773 
6.756 
6.740 

6.724 
6.707 
6.691 
6.674 
6.658 
6.641 

6.625 
6.608 
6.591 
6.575 
6.558 
6.541 

6.524 
6.507 
6.490 
6.472 
6.455 
6.438 

6.421 
6.403 
6.386 
6.363 
6.351 
6.333 

Inches. 
9.584 
9.565 
9.545 
9.526 
9.506 
9.486 

9.466 
9.446 
9.  426 
9.406 
9.386 
9.366 

9.345 
9.325 
9.304 
9.284 
9.263 
9.242 

9!  200 
9.179 
9.158 
9.137 
9.116 

9.095 
9,073 
9.052 
9.030 
9.008 
8.987 

8.965 
8.  943 
8.921 
8.899 
8.877 
8.855 

8.834 
8.811 
8.788 
8.766 
8.744 
8.721 

8.698 
8.676 
8.653 
8.630 
8.607 
8.584 

8.661 
8.538 
8.514 
8.491 
8.468 
8.444 

Inches, 
11.  980 
11.  956 
11.  932 
11.  907 
11.  883 
11.  858 

11.  833 
11.  808 
11.  783 
11.  757 
11.  732 
11.  707 

11.  682 
11.  656 
11.630 
11.  605 
11.  579 
11.  553 

11.  527 
11.501 
11.  474 
11.  448 
11.  421 
11. 395 

11.  309 
11.  342 
11.  315 
11.  288 
11.  261 
11.  234 

11.207 
11. 179 
11.  152 
11. 124 
11.  097 
11.069 

11.  042 
11.014 
10.  985 
10.  958 
10.  929 
10.  901 

10.  873 
10.  844 
10. 816 
10.  787 
10.  759 
10.  730 

10. 701 
10.  072 
10.  643 
10.  614 
10.  585 
10.  556 

Inches. 
14.  376 
14.  347 
14.  318 
14.  288 
14.  259 
14.  230 

14.  200 
14. 170 
14. 139 
14. 109 
14.  078 
14.048 

14.  018 
13.  987 
13.  956 
13.  925 
13.  894 
13.  864 

13.832 
13.  801 
13.  769 
13.  737 
13.  705 
13.  673 

13.  642 
13.  610 
13.  577 
13.  545 
13.  513 
13.  480 

13.448 
13. 415 
13.  382- 
13.  349 
13.  316 
13.  283 

13.  250 
13.217 
13. 183 
13. 149 
13. 115 
13.  081 

13.  048 
13.  013 
12.  979 
12.  945 
12.  910 
12.  876 

12.  842 
12.  807 
12.  772 
12.  737 
12.  701 
12.  667 

|1 
0  a 

34° 

35° 

5. 824 
11.  647 
17.  471 
23.294 
.  29.118 

5 
10 
15 
20 
25 
30 

Inch. 

0.001 
.004 
.009 
.016 
.025 
.036 

Inch. 

0.001 
.004 
.009 
.016 
.025 
.036 

5,824 
11.  649 
17. 473 
23.  297 
29. 122 

36= 

37° 

5.826 
11.651 
17.477 
23.  302 
29. 128 

5 

10 
15 
20 
25 
30 

O.OOI 
.004 
.009 
.016 
.025 
.036 

0.001 
.004 
.009 
.016 
.026 
.037 

5.827 
U.  653 
17. 480 
23. 306 
29. 133 

37°    ■ 

38° 

5 

10 
15 
20 
25 
30 

Inch. 

0.001 
.004 
.009 
.016 
.026 
.037 

Inch. 
0.001 
.004 
.009 
.017 
.026 
.037 

5.828 
11.  655 
17.  483 
23.  310 
29. 138 

5.829 
11.  657 
17.486 
23.  314 
29. 143 

39° 

40° 

6 
10 
15 
20 
25 
30 

0.001 
.004 
.009 
.017 
.026 
.037 

0.001 
.004 
.009 
.017 
.026 
.038 

5.830 
11.  659 
17. 489 
23.319 
29. 149 

40° 

41° 

5.831 
11.  661 
17.  492 
23.  323 
29.154 

5 
10 
15 
20 
25 
30 

Inch. 

0.001 
.004 
.009 
.017 
.026 
.038 

Inch. 

0.001 
.004 
.009 
.017 
.026 
.038 

5.832 
11.  663 
17.  495 
23.  327 
29. 159 

42° 

43° 

5 
10 
16 
20 
25 
30 

0.001 
.004 
.010 
.017 
.026 
.038 

0.001 
.004 
.010 
.017 
.027 
.038 

1 

PEOJECTION  TABLES. 


179 


Table  XXI. — Coordinates  for  projection  of  maps.    Scale  - 
[Prepared  by  E.  S.  Woodwaxd.] 


-Coutinued. 


3 

l3 

i 

Pi 

•ggg§ 

■3 

Abscissas  of  developed  parallel. 

Ordinatea  of  devel- 
oped parallel. 

5'  longi- 
tude. 

10'  longi- 
tude. 

15'  longi- 
tude. 

20'  longi- 
tude. 

25'  longi- 
tude. 

30'  longi- 
tude. 

44 

46 
47 
48 
49 
60 

00 
10 
20 
30 
40 
50 

00 
10 
20 
3Q 
40 
50 

00 
10 
20 
30 
40 
50 

00 
10 
20 
30 
40 
50 

00 
10 
20 
30 
40 
50 

00 
10 
20 
30 
40 
50 

00 

Inches. 

Inches. 
2.105 
2.099 
2.093 
2.087 
2.081 
2.076 

2.070 
2.064 
2.057 
2.051 
2.045 
2.039 

2.033 
2.027 
2.021 
2.015 
2.009 
2.003 

1.996 
1.990 
1.984 
1.978 
1.971 
1.965 

1.959 
1.952 
1.946 
1.940 
1.933 
1.927 

1.921 

1.914 
1.908 
1.901 
1.895 
1.888 

1.882 

Inches. 
4.210 

■    4.199 
4.187 
4. 175 
4.163 
4.151 

4.139 
4.127 
4115 
4.103 
4.091 
4.079 

4.067 
4.054 
4.042 
4.030 
4.017 
4.005 

3.992 
3.980 
3.968 
3.955 
3.943 
3.930 

3.917 
3.905 
8.892 
3.879 
3.867 
3.854 

3.841 
3.828 
3.815 
3.803 
3.790 
3.777 

3.764 

Inches. 
6.316 
6.298 
6.280 
6.262 
6.244 
6.227 

6.209 
6.191 
6.172 
6.154 
6.136 
6.118 

6.100 
6.081 
6.063 
6.044 
6.026 
6.008 

5.989 
5.  970 
5.951 
5.933 
5.914 
5.895 

5.876 
5.857 
5.838 
6.819 
5.800 
6.781 

5.762 
5.743 
5.723 
5.704 
5.684 
6.665 

5.646 

Inches. 
8.421 
8.397 
8.373 
8.350 
8.326 
8.302 

8.278 
8.264 
8.230 
8.206 
8.181 
8.157 

8.133 
8.108 
8.084 
8.069 
8.034 
8.010 

7.985 
7.960 
7.935 
7.910 
7.885 
7.860 

7.835 
7.810 
7.784 
7.769 
7.733 
7.708 

7.682 
7.657 
7.631 
7.605 
7.579 
7.563 

7.527 

Inches. 
10. 526 
10.  496 
10.  467 
10. 437 
10.407 
10.  378 

10.343 
10.  317 
10.  288 
10.  257 
10.  226 
10. 197 

10. 166 
10. 136 
10. 104 
10.074 
10.043 
10. 013 

9.981 
9.951 
9.919 
9.888 
9.857 
9.826 

9.794 
9.762 
9.730 
9.699 
9.667 
9.635 

9.603 
9.571 
9.539 
9.507 
9.174 
9.442 

9.409 

Inches. 
12.  631 
12.  596 
12.  560 
12.524 
12.489 
12.453 

12.417 
12.381 
12.345 
12.  308 
12.  272 
12.  236 

12. 199 
12. 163 
12. 125 
12.  089 
12.052 
12.  015 

11.  978 
11.  941 
11.  903 
11.866 
11.  828 
11.  791 

11.  752 
11. 714 
11.  677 
11.  638 
U.600 
11.  562 

11.523 
11.485 
11. 446 
11.  408 
11.  369 
11.330 

11.291 

Mo 

(3- 

43" 

44° 

5.833 
11.666 
17.  498 
23.331 
29. 164 

5 

10 
15 
20 
25 
30 

Inch. 

0.001 
.004 
.010 
.017 
.027 
.038 

Inch. 
0.001 
-.004 
.010 
.017 
.027 
.038 

5.834 
11.668 
17.  501 
23.335 
29. 169 

45° 

46° 

5.835 
11.670 
17.  504 
23.  339 
29. 174 

5 
10 
15 
20 
25 
30 

0.001 
.004 
.010 
.017 
.027 
.038 

0.001 
.004 
.010 
.017 
.027 
.038 

5.  836 
11.  672 
17.  508 
23.344 
29.180 

470 

48° 

5 
10 
15 
20 
25 
30 

0.001 
.004 
.010 
.017 
.027 
.038 

0.001 
.004 
.010 
.017 
.026 
.038 

5.837 
11.674 
17.511 
23.348 
29. 185 

5.838 
11.  676 
17.  514 
23.352 
29. 190 

490 

50° 

5 
10 
15 
20 
25 
30 

0.001 
.004 
.010 
.017 
.026 
.038 

0.001 
.004 
.009 
.017 
.026 
.038 

1 

180 


A  MANUAL  OF  TOPOGKAPHIC  METHODS. 


Table  XXH.- 


-Coordinates  for  projection  of  maps.     Scale  ^j-suxr- 
[Prepared  by  R.  S.  "Woodward.] 


11 

Hi 

So 
llli 

3.1 1* 

Absci 

>sa3  of  developed  parallel. 

Ordinates  of  devel- 
oped parallel. 

2J'  longi- 
tude. 

5'  loiiffi- 
tude. 

7J'  longi- 
tude. 

lOMongi- 
tude. 

12J' lon- 
gitude. 

15'  longi- 
tude. 

25  00 
05 
10 
15 
20 
25 
30 

il 

45 
50 
55 

26  00 
05 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

27  00 
05 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

28  00 
05 
10 

15 
20 
25 
30 
35 
40 
45 
50 
55 

29  00 
05 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

30  00 
05 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

Inches. 

Inches. 
2. 650 
2.648 
2.646 
2.644 
2.642 
2.641 
2.639 
2.637 
2.635 
2.  633 
2.631 
2.630 

2.628 
2.626 
2.624 
2.622 
2.620 
2.618 
2.617 
2.615 
2.613 
2.611 
2.609 
2.607 

2.605 
2.603 
2.601 
2.599 
2.597 
2.595 
2.593 
2.591 
2.590 
2.588 
2.586 
2.584 

2.582 
2.580 
2.578 
2.576 
2.574 
2.572 
2.570 
2.568 
2.566 
2.564 
2.562 
2.360 

2.558 
2.555 
2.553 
2.551 
2.549 
2.547 
2.545 
2.543 
•    2.541 
2.539 
2.337 
2.535 

2.533 
2.530 
2.528 
2.526 
2.524 
2.522 
2.520 
2.518 
2.515 
2.513 
2.511 
2.509 

Inches. 
5.299 
5.296 
5.292 
5.288 
5.285 
5.281 
5.277 
5.274 
5.270 
5.266 
5.263 
5.259 

5.256 
5.252 
5.248 
5.244 
5.241 
5.237 
5,233 
5.  229 
5.225 
5.222 
5.218 
5.214 

5.210 
5.207 
5.203 
5.399 
5.195 
5.191 
5.187 
5.183 
5.179 
5.175 
5.171 
5. 167 

5.163 
5.159 
3.155 
5.151 
5.147 
5.143 
5.139 
5.135 
5.131 
5.127 
5.123 
5.119 

5.115 
5.111 
5.107 
5.103 
5.098 
5.094 
5.090 
5.086 
5.082 
5.078 
5.073 
5.069 

5.065 
5.061 
5.057 
5.052 
5.048 
5.044 
5.039 
5.035 
5.031 
5.026 
5.022 
5.018 

Inches. 
7.949 
7.944 
7.938 
7.933 
7.927 
7.922 
7.916 
7.911 
7.905 
7.900 
7.894 
7.689 

7.883 
7.878 
7.872 
7.866 
7.861 
7.853 
7.849 
7.844 
7.838 
7.833 
7.827 
7.821 

7.816 
7.810 
7.804 
7.798 
7.792 
7.786 
7.780 
7.774 
7.769 
7.763 
7.757 
7.751 

7.745 
7.739 
7.733 
7.727 
7.721 
7.715 
7.709 
7.703 
7.697 
7.691 
7.685 
7.679 

7.673 
7.666 
7.660 
7.654 
7.648 
7.641 
7.635 
7.629 
7.623 
7.616 
7.610 
.     7. 604 

7.598 
7.591 
7.583 
7.578 
7.572 
7.565 
7.359 
7.552 
7.546 
7.540 
7.533 
7.527 

Inches. 
10.  399 
10.  591 
10.  584 
10.  577 
10.  569 
10.  562 
10.  553 
10.548 
10.  540 
10.  533 
10.  526 
10.  518 

10.511 
10.  504 
10. 496 
10.  489 
10.  481 
10. 473 
10.  466 
10.  458 
10.451 
10.443 
10.436 
10.  428 

10.  421 
10.413 
10. 405 
10.  397 
10.  389 
10.  382 
10.  374 
10.  366 
10.  358 
10.  330 
10.  342 
10.333 

10.  327 
10.  319 
10.  311 
10.  303 
10.  294 
10.  286 
10.  278 
10.  270 
10.  262 
10.  234 
10.  246 
10.  238 

10.  230 
10.  222 
10.  213 
10.  205 
10.197 
10. 188 
10. 180 
10. 172 
10.164 
10. 133 
10. 147 
10. 138 

10. 130 
10. 122 
10. 113 
10. 104 
10.  096 
10.  087 
10.  079 
10.  070 
10.  061 
10.  053 
10.044 
10.  036 

Inches. 
13.248 
13.  239 
13.230 
13.  221 
13.  212 
13.  203 
13. 194 
13.184 
13. 175 
13. 166 
13.157 
13. 148 

13. 139 
13. 129 
13. 120 
13.  Ill 
13. 101 
13.  092 
13.  082 
13.  073 
13.064 
13.034 
13.  045 
13.  035 

13.  026 
13.  016 
13. 006 
12.  997 
12.  987 
12.  077 
12.  967 
12.  937 
12.  948 
12.  938 
12.  928 
12.  918 

12.  908 
12.  898 
12.  888 
12.  878 
12.  868 
12.  858 
12.  848 
12.  838 
12.  828 
12.  818 
12.  808 
12.  798 

12.  788 
12.  777 
12.  767 
12.  756 
12.  746 
12.  735 
12.  725 
12.  716 
12.  704 
12.  694 
12. 684 
12.  673 

12.  663 
1.2.  652 
12.  641 
12.  630 
12.  620 
12.  609 
12.  598 
12.587 
12.  577 
12.  566 
12.  355 
12.544 

Inches. 
15. 898 
15.  887 
13.  876 
15.  865 
15.  854 
15.  843 
15.  832 
15.  821 
13.  810 
15.  799 
15.  788 
15. 777 

15.  766 
15.  755 
13.744 
15.733 
13.  721 
15.  710 
15.  699 
15.  688 
15.  676 
15.  665 
15.  654 
15.  642 

15.  631 
15.  620 
15.  608 
15.  596 
15.  584 
15.  572 
15.  561 
15.  549 
15.  537 
15.  325 
15.  514 
15.  502 

15.  490 
13.478 
15.  466 
15.454 
15.  442 
15.  430 
15.  418 
15.  403 
15.  393 
15.  381 
15.  369 
15.  357 

15.  345 
15.  333 
15.  320 
15.  308 
15.  295 
15.  283 
15.  270 
15.  258 
15. 245 
15.  233 
15.  220 
15.  208 

15. 195 
15. 182 
15. 169 
15. 157 
15. 144 
15. 131 
15. 118 
15. 105 
13.  092 
15.  079 
15.  066 
15.  053 

25° 

26° 

5.815 
11.  629 
.     17. 444 
23.  259 
29.  074 
34. 888 

24 
5 

1? 

15 

Inch. 

0.000 
.002 
.004 
.007 
.010 
,015 

Inch. 

0,000 
.002 
.004 
.007 
.010 

•  .015 

1 

5.816 
.    11.631 
17.447 
23.  262 
29.  078 
34.893 

27° 

2* 
5' 
74 
10 
12i 
15" 

Inch. 

0.000 
.002 
.004 
.007 
.011 
.015 

5.816 
11.  633 
17.449 
2.3.  265 
29.  082 
34.898 

1 

27° 

.28° 

f 

1? 
12i 
15 

Inch. 

0.000 
.002 
.004 
,007 
.011 
.013 

Inch. 

0.000 
.002 
.004 
.007 
.011 
,016 

5.817 
11.634 
17.  451 
23.  268 
29. 085 
34.  903 

1 

29° 

f 
1? 
If 

Inch. 

0.000 
.002 
.004 
.007 
.011 
.016 

5.818 
11.  636 
17.  434 
23.  272 
29.  090 
34.  908 

1 

29° 

30° 

2* 
3 

1? 
124 
15 

Inch. 

0.000 
.002 
.004 
.007 
.011 
.016 

Inch. 

0.000 
.002 
.004 
.007 
.012 
.017 

5.819 
11.  638 
17.  457 
23.  276 
29.  095 
34.913 

PEOJECTION  TABLES. 


181 


Table  XXll.—Coardinates  for  projection  of  maps.     Scale  -^ji^, — Continued. 
[Prepared  by  E.  S.  "Woodward.] 


Inches. 

'"'s.'sm' 

11.  640 
17.  460 
23.  280 
29. 100 
34.  919 


5.821 
11.645! 
17.462 
23.  283 
29. 104 
34.  925 


5.822 
11.643 
17. 465 
23.  287 
29. 109 
34.  930 


5.823 
11.  645 
17.468 
23.  291 
29. 113 
34.  936 


5.824 
11.  647 
17. 471 
23.  294 
29. 118 
34.  942 


5.824 
11.649 
17. 473 
23.  297 
29. 122 
34.  946 


Abscissas  of  developed  parallel. 


Inches. 

.  2.  507 
2.505 
2.502 
2.500 
2.498 
2.496 
2.494 
2.491 
2.489 
2.487 
2.485 
2.482 


2.478 
2.476 
2.473 
2.471 
2.469 
2.407 
2.464 
2.462 
2.460 
2.458 
2.455 

2.453 
2.451 
2.448 
2.446 
2.444 
2.441 
2.439 
2.437 
2.434 
2.432 
2.430 
2.427 

2.425 
2.423 
2.420 
2.418 
2.415 
2.413 
2.411 
2.408 
2.406 
2.403 
2.401 


2.384 
2.381 
2.379 
2.376 
2.374 
2.372 
2,369 

2.367 
2.364 
2.362 
2.359 
2.357 
2.354 
2.352 
2.349 
2.346 
2.344 
2.341 


Inches. 
5.014 
5.009 
5.005 
5,000 
4.996 
4.991 
4.987 
4.983 
4.978 
4.974 


4.956 
4.951 
4.947 
4.942 


4.924 
4.920 
4.915 
4.910 


4.850 
4.845 
4.840 


4.797 

4.792 
4.787 
4.782 
4.777 
4.773 
4.768 
4.763 
4.758 
4.753 
4.748 
4.743 
4.738 

4.733 
4.728 
4.723 
4.718 
4.713 
4.708 
4.703 
4.698 
4.693 


Inches. 
7.520 
7.514 
7.507 
7.5U0 
7.494 
7. 487 
7.480 
7.474 
7.467 
7.460 
7.454 
7.4A7 

7.441 
7.434 
7.427 
7.420 
7.413 
7.407 
7.400 
7.393 
7.  3S6 
7.379 
7,372 
7.366 


7.331 
7.324 
7.317 


7.303 

7.296 

7.289 

7.282 

7.275 

7.267 

7.260 

7.253 

7.246 

7.239 

7.231 

7.224 

7.217 

7.210 

7.203 

7.195 

7.188 

7.181 

7.174 

7.166 

7.159 

7.151 

7.144 

7.137 

7.129 

7.122 

7.115    ' 

7.107 

7.100 

7.092 

7.085 

7.077 

7.070 

7.062 

7.  055 

7.047 

7.039 

7.032 

7.024 

7.  017 

Inches. 
10.  027 
10.  018 
10.  009 
10.  000 


9.974 
9.965 
9.956 
9.947 


9. 793 
9.784 
9.774 


9.728 
9,718 
9.709 

9.700 
9.690 
9.680 
9.671 
9,661 
9.652 
9.642 
9.632 
9.  623 
9.613 
9.604 
9.594 

9.  584 
9.  574 
9.565 
9.555 
9.545 
9.535 
9.  525 
9.516 
9.506 
9.496 
9.486 
9.476 

9.466 
9.456 
9.446 
9.436 
9.426 
9.416 
9.406 
9.396 
9.380 
9.376 
9.366 
9.356 


Inches. 
12. 534 
12.  523 
12.  512 
12.  500 
12.  489 
12.  478 
12.  467 
12.  456 
12.445 
12. 434 
12. 423 
12. 412 

12. 401 
12.  390 
12.  378 
12.  367 
12.  356 
12.344 
12.  333 
12.  322 
12.  310 
12.  299 
12.  287 
12.  276 

12.  265 
12.  253 
12.  241 
12.  230 
12.  218 
12.  206 
12. 195 
12. 183 
12. 171 
12. 160 
12.148 
12. 136 

12.124 
12. 112 
12. 100 
12.  088 
12.  076 
12.  064 
12.  052 
12.  040 
12.  028 
12.016 
12.  004 
11. 992 

11.  980 
11. 908 
11.  956 
11.  914 
11.  931 
11.  919 
11.  907 
11.  895 
11.882 
II.  870 
11.858 
11.  845 

11.833 
11.  820 
11.  808 
11.  795 
11.  783 
11.770 
11.  758 
11.  745 
11.  732 
11.  720 
11.  707 
11.694 


Inches. 
15. 040 
15.  027 
15.  014 
15.  000 
14.  987 
14.  974 
14. 961 
14.  948 
14.  934 
14.  921 
14.908 
14. 894 

14.  881 
14,  868 
14.  854 
14.  840 
14.  827 
14. 813 
14.  800 
14.  786 
14.  772 
14.  759 
14.  745 
14.  731 


14.  690 
14.  676 
14.  662 
14.  648 
14.  633 
14.  619 
14.'  605 
14.  591 
14.  577 
14.  563 

14.  549 
14.  535 
14.  520 
14.  506 
14.  492 
14.  477 
14.  463 
14. 448 
14. 434 
14.  420 
14. 405 
14.  391 

14. 376 
14.  362 
14.347 
14.  332 
14. 318 
14.  303 
14.  288 
14.  273 
14.  259 
14.244 
14.  229 
14.  214 

14.  200 
14. 183 
14. 169 
14, 154 
14, 139 
14. 124 
14. 109 
14.  094 
14.  079 
14.  064 
14,  048 
14,  033 


Ordinates  of  devel- 
oped parallel. 


Inch. 
0.000 
.002 


Inch. 
0.000 
.002 


182 


A  MANUAL  OF  TOPOGKAPHIC  METHODS. 


Table  XXII. — Coordinates  for  ^projection  of  maps.     Scale -^xm — Coutinued. 
[Prepared  Ijy  E.  S.  'Woodwartl,] 


Abscissas  of  developed  parallel. 


12J'  lon- 
gitude. 


Ordiilatea  of  devel- 
oped parallel. 


Inches. 


5.826 
11.651 
17.  477 
23.  302 
29. 128 
34.  954 


5.  828 
11.655 
17. 483 
23.  310 
29. 138 
34. 966 


5.829 
11.  657 
17.  486 
23.  314 
29. 143 
34.  972 


5.830 
11.  659 
17.489 
23.319 
29. 149 
34.  978 


5.831 
11.  661 
17.  492 
23.  323 
29. 154 
34.984 


2.  323 
2.321 
2.318 
2.  316 
2.313 
2.311 
2.308 


2.300 
2.  298 
2.295 
2.292 
2.290 
2.287 
2.284 


1.  282 


2.274 
2.271 
2.268 
2.266 


2.250 
2.247 
2.244 

2.241 
2.239 
2.236 
2.233 
2.230 
2.228 


2.211 

2.208 
2.206 
2.203 
2.200 
2.197 
2.194 
2.192 
2.189 
2.186 
2.183 
2.180 
2.178 

2.175 
2.172 
2.169 
2.166 
2.163 
2.160 
2.158 
2.155 
2.152 
2.149 
2.  146 
2.143 


Inches. 
4.673 
4.667 
4.662 
4.657 
4.652 
4.647 
4.642 
4.637 
4.631 
4.626 
4.621 
4.616 

4.611 
4.606 
4.600 
4.595 
4.590 


4.547 
4.542 
4.537 
4.531 
4.526 
4.521 
4.515 
4.510 
4.504 
4.499 
4.494 


4.472 
4.466 
4.461 
4.455 
4.450 
4.444 
4.439 
4.433 
4.428 
4.422 

4.417 
4.411 
4.406 
4.400 
4.394 
4.389 
4.383 
4.377 
4.372 


4.349 
4.344 
4.338 
4.332 
4.326 
4.321 
4.315 


6.  963 
6.955 
6.947 
6.939 
6.  932 
6.924 

6.916 

6.908 
6.900 
6.892 
6.885 
6.877 
6.869 
6.861 
6.853 
6.845 
6.837 


6.821 
6.813 
6.805 
6.797 
6.789 
6.781 
6.773 
6.765 
6.757 
6.748 
6.740 
6.732 

6.724 
6.716 
6.708 
6.699 
6.691 
6.683 
6.675 
6.666 
6.658 
6.050 
6.642 
6.633 


6.600 
6.591 
6.583 
6.575 
6.666 
6.558 
6.549 
6.541 
6.533 

6.524 
6.515 
6.507 
6.498 
6.490 
6.481 
6.472 
6.464 
6.455 
6.447 
6.438 
6.429 


Inches. 
9.345 
9.335 
9.325 
9.314 
9.304 
9.294 
9.283 
9.273 
9.363 
9.253 
9.212 
9.233 

9.222 
9.211 
9.201 
9.190 
9.179 
9.169 
9.158 
9.148 
9.137 
9.137 
9.118 
9.106 


9.095 
9.084 
9.073 
9.063 
9.052 
9.041 
9.030 
9.020 
9.009 
8.998 
8.987 


Inches. 
11.  683 
11.  669 
11.  656 
11.643 
11.630 
11.617 
11.  604 
11.  591 
11.  578 
11.566 
11.  553 
11.  540 

11.527 
11.  514 
11.  501 
11.  488 
■11.474 
11.461 
11.448 
11.  435 
11.422 
11.  408 
11.  395 
11.  382 

11.  369 
11.  355 
11.  342 
11.  328 
11.315 
11.301 
11.288 
11.  374 
11.261 
11.  247 
11.234 
11.221 

11.  207 
11. 193 
11. 180 
11. 166 
11. 152 
11.  138 
11.124 
14. Ill 
II.  097 
11.  083 
11.069 
11.  056 

11.042 
11.  038 
11.014 
11.000 
10.  986 
10.  972 
10.  958 
10.  944 
10.  930 
10.  916 
10.  902 
10.  888 

10.  873 
10.  859 
10.845 
10.  830 
10.816 
10.  803 
10.  787 
10. 773 
10.  759 
10.744 
10.  730 
10.  716 


14.  003 
13.  987 
13.  972 
13,956 
13.941 
13.925 
13.910 
13.  894 
13.  879 
13.  863 
13.  848 

13.  832 
13.  817 
13.  801 
13.  785 
13.  769 
13.  753 
13.  737 
13.  722 
13.  706 
13.  690 
13.  674 
13.  058 

13.  642 
13.  626 
13.  610 
13.  594 
13. 578 
13.  562 
13.  545 
13.529 
13.  513 
13.  497 
13.  481 
13. 465 

13.448 
13. 432 
13. 415 
13.  399 
13.  382 
13.360 
13.  349 
13. 333 
13.  316 
13.  300 
13.  283 
13.  267 

13.  250 
13.  233 
13.  216 
13.  200 
13. 183 
13. 166 
13. 149 
13. 133 
13. 115 
13.  099 
13.  082 
13.  065 


13.  014 
13.  996 
12.  979 
12.  963 
12.945 
12.  928 

12.  910 

13.  893 
12.  876 
12.  859 


40° 

Inch. 

2i 

0. 001 

5 

.002 

u 

.005 

0 

.008 

2.V 

.013 

.019 

41° 

Inch. 

'i>, 

0.001 

5 

.002 

7S: 

.005 

10 

.008 

12* 

.013 

15 

.019 

PEOJECTIOI^r  TABLES. 


183 


Table  XXll.— Coordinates  for  projection  of  maps.     Scale  ^jnr — Continued. 
[Prepared  by  E.  S.  "Woodward.] 


Inches. 

'""5.'832' 
11.  663 
17.  495 
23.  327 
29.  159 


Absci9aa.s  of  developed  parallel. 


23.  331 
29. 164 
34.  997 


5.834 
11.  668 
17.  501 
23.  335 
29. 169 
35.  003 


5.835 
11.  670 
17.  504 
23.  339 
29. 174 
36.  009 


5.836 
11.  672 
17.  508 
23.  344 
29.  ISO 
35. 015 


5.837 
11.  674 
17.511 
23.  348 
29. 185 
35.  021 


Inches. 
2.140 
2.137 
2.134 
2.132 
2.129 
2.126 
2.123 
2.120 
2.117 
2.114 
2.111 
2.108 


2.096 
2.093 
2.090 
2.087 
2.084 
2.C81 
2.078 
2.076 
2.073 


2.  045 
2.042 
2,039 
2.036 


1.996 
1.993 
1.990 
1.987 
1.984 
1.981 
1.977 
1.974 
1.971 
1.968 
1.965 


1.959 
1.956 
1.952 
1.949 
1.946 
1.943 
1.940 
1.937 
1.933 
1.930 
1.927 
1.924 


4.275 
4.269 
4.263 
4.257 
4.251 
4.246 
4.240 
4.234 
4.228 
4.222 
4.216 


4.193 
4.187 
4.181 
4.175 
4.169 
4.163 
4.157 
4.151 
4.145 


4.133 
4.127 
4.121 
4.115 
4.109 
4.103 
4.097 
4.091 
4,085 
4.079 
4.073 

4.067 
4.060 
4.054 
4.048 
4.042 
4.036 
4.030 
4.023 
4.017 
4.011 
4.005 


3,917 
3.911 
3.905 


Inches. 
6.421 
6.412 
6.403 
6.395 
6.386 
6.377 


6.351 
6.342 
6.333 
6.324 

6.316 
6.307 
6.298 


6.271 
6.262 
6.253 
6.244 
6.235 
6.227 
6.  218 

6.209 
6.200 
6.191 
6.181 
6. 172 
6.163 
6.154 
6.145 
6.136 
6.127 
6.118 
6.109 

6.100 
6.091 
6.081 
6.072 
6.063 
6.054 
6.044 
6.035 
6.026 
6.017 
6.008 


5.970 
5.981 
5  951 
5.942 
5.933 
5.923 
5.914 
5.904 
5.895 


5.876 
5-.  867 
5.857 
5.848 


Inches. 
8.561 
8.550 


8.503 
8.491 
8.479 
8.468 
8.456 
8.444 
8.432 


8.326 
8.314 
8.  302 . 
8.290 

8.278 
8.266 
8.254 
§.242 
8.230 
8.218 
8.206 
8.194 
8.181 
8.169 
8.157 


.145 


7.985 
7.973 
7.960 
7.948 
7.935 
7.923 
7.910 
7.898 


7.784 
7.771 
7.759 
7.746 
7.733 
7.721 
7.708 
7.695 


Inches. 
10.  701 
10.  687 
10.  672 
10.  658 
10. 643 
10.  628 
10.  614 
10.  599 
10.  585 
10.  570 
10. 555 
10.541 


10. 496 
10.482 
10.  467 
10.  452 
10.437 
10.  422 
10.  407 


10.; 


10. 348 
10.  333 
10.  318 
10.  302 
10.  287 
10,  272 
10.  257 
10.  242 
10.  227 
10.  212 
10. 197 
10. 182 

10. 166 
10. 151 
10. 136 
10. 120 
10. 105 
10.  090 
10.  074 
10.  059 
10.  043 
10.  028 
10.  013 
9.907 

9.982 
9.966 
9.950 
9.935 
9.919 


9.667 
9.651 
9.635 
9.619 


Inches. 
12.  842 
12. 824 
12.  807 
12.  789 
12.  772 
12.  754 
12.  736 
12. 719 
12.  701 
12.  684 
12.  666 
12. 649 

12.  631 
12.  613 
12.  596 
12.  578 
12.  560 
12.  542 
12.524 
12.  506 
12.  489 
12.  471 
12.  453 
12.  43S 

12. 417 
12.  399 
12.  381 
12.  363 
12.  345 
12.  327 
12.  308 
12.  290 
12.  272 
12.  254 
12. 236 
12.  218 

12.  200 
12. 181 
12. 163 
12. 144 
12. 126 
12. 107 
12.  089 
12.  070 
12.  052 
12.  033 
12.015 
11.  996 

II.  978 
11.  959 
11.  940 
11. 922 
11.  903 
11.  884 
11.865 
11.  846 
11.  828 
11.  809 
11.  790 
11.  771 

11.  752 
11.733' 
11.  714 
11.  695 
11.  676 
11.  657 
11.  638 
11. 619 
11.  600 
11.  581 
11.  .562 
11.  543 


Ordinates  of  devel- 
oped parallel. 


184 


A  MANUAL  OF  TOrOGEAPHIC  METHODS. 


Table  XXII. — Coordinates  for  projection  of  maps.     Scale -g^jy^ 
[Prepared  hy  E.  S.  Woodward.] 


■3.S£ 


Inches. 


5.838 
11.  676 
17. 51J, 
23.  352 
29. 190 
35.  027 


Abscissas  of  developed  parallel. 


JJ'  longi-    5'  longi,    7^'  longi-   10'  longi-    12J'  Ion-    15'  long 
tude.  tilde.  tude.  tude.        gitude.        tude. 


Inches. 
1.921 
1.917 
1.9U 
1.911 


3.828 
3.822 
3.815 
3.809 
3.802 
3.796 
3.790 
3.783 
3.777 


5.742 
5.733 
5.723 
5.713 
5.704 
5.694 
5.684 
5.675 
5.665 
5.655 


7.670 
7.657 
7.644 
7.631 
7.618 
7.605 
7.592 
7.579 
7.566 
7.553 
7.540 

7.528 


Inches. 
9.003 
9.587 
9.571 
9.555 
9.  538 
9.  522 
9.506 
9.490 
9.474 
9.458 
9.442 
9.426 


9.409 


Inches. 
11.  524 
11.  504 
11.485 
11. 466 
11.446 
11.427 
11.  407 
11.  388 
11.  369 
11.  349 
11.  330 
11.  311 


Ordinates  of  devel- 
oped parallel. 


49° 

"^ 

Inch. 

^ 

0.001 

5 

.002 

74 
0 

.005 
.008 

n 

.013 

5 

.039 

Table  XXIII. — Coordinates  for  ))rojecUon  of  maps.     Scale  -^-^^jfij. 
[Prepared  by  S.  S.  Gannett.] 


Latitade 
par.allel. 

Abscissas  of  developed  p.irallel. 

Ordinates  of  devel- 
oped parallel. 

Longitude  interval. 

Longi- 

5' 

7i' 

10' 

15' 

tude 

Incb. 

interval. 

"  o 

Inches. 

Inches. 

Inches. 

Inches. 

, 

39    00 

6.316 

9.474 

12.632 

18.948 

5 

.003 

05 

.309 

.463 

.617 

.926 

n 

.007 

07i 

.305 

.457 

.609 

.914 

10 

.012 

10 
15 

.301 
.294 

.451 
.440 

.602 
.587 

.903 
.881 

15 

.026 

20 

6.286 

9.429 

12.572 

18.  858 

Latitude 
interval. 

Meridi- 
onal dis- 

22* 
25 

.282 
.279 

.423 
.418 

.565 
.557 

.847 
.836 

/ 

Inch. 

30 

.271 

.406 

.542 

.813 

1 
2 

1.619 
3.237 

35 

6.264 

9.395 

12.  527 

18.  791 

4 
6 
6 
7 

0.475 
8.094 
9.712 
11. 331 

37J 

.260 

.389 

.520 

.780 

40 

.256 

.384 

.512 

.768 

45 

.J49 

■.873 

.497 

.746 

8 

12.  960 

50 

6.2a 

9.361 

12.482 

18.  723 

9 

14.  569 

52i 
65 

.237 
.234 

.356 
.350 

.475 
.467 

.712 
.701 

10 

16. 188 

60 

.226 

.339 

.452 

.678 

Longi- 
tude in- 

Inch. 

40    00 
05 

6.226 
.219 

9.339 
.328 

12.  452 
.438 

18.  678 
.656 

' 

07i 

.215 

.322 

.429 

.644 

5 

.003 

10 

.211 

.316 

.422 

.633 

n 

.007 

15 

.203 

.305 

.406 

.609 

10 

.012 

20 

6.196 

9.293 

12.  392 

18.  587 

15 

.026 

22J 

.192 

.288 

.384 

.576 

Latitude 

25 

.188 

.282 

.376 

.564 

30 

.180 

.270 

.361 

.540 

' 

Inch. 

35 

6.173 

9.259 

12.  346 

18. 518 

1 
2 
3 
4 

1.619 
3.238 
4.857 
6.470 
8.095 

37* 

.169 

.253 

.338 

.506 

40 

.165 

.247 

.330 

.495 

45 

.157 

.236 

.315 

.472 

6 

9.714 

50 

6.150 

9.224 

12.300 

18.449 

7 

11.  333 

52J 

.146 

.219 

.292 

.438 

8 

12.  952 

55- 

.142 

.213 

.285 

.427 

9 

14.  571 

60 

.134 

.201 

.269 

.403 

10 

16. 190 

1 

PEOJEOTION  TABLES. 


185 


Table  XXIII. — Coordinates  for  projection  of  maps.     Scale  4-5^513 — Continued. 
(Prepared  by  S.  S.  Gannett.] 


Abscissas  of  developed  parallel. 


Longitude  interval. 


Ordinates  of  devel- 
oped parallel. 

Loncfi- 
tnde  Inch, 

interval. 


Meridi- 
onal dis- 
tance. 

1.619 
3.239 
4.858 
6.477 
8.097 
9.716 
11.335 
12.  955 
14.  574 
16. 193 


.074 
.051 


■*B.961 


18.  027 
.015 
.003 

17.  979 

17. 956 
.944 
.933 


Meridi- 
onal dis- 
tance. 


6.478 
8.098 
9.718 

11.  337 

12.  957 
14.  576 
16. 196 


Longi- 
tude in-       lucli. 
terval. 


Meridi- 
onal dis- 
tance. 

Inch. 
1.620 
3.  2.J0 


186 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXIII. — Coordinates  for  projection  of  maps.     Scale  tsW — Continued. 
[Prepared  by  S.  S.  Gannett.] 


Latitude 

of 
Iparallel. 

Abscissas  of  developed  parallel. 

Ordinates  of  devel- 
oped parallel. 

Longitude  interval. 

Longi- 

6' 

7J' 

10' 

15' 

tude 
interval. 

Tncb. 

0             1 

Inches. 

Inches. 

Inches. 

Inches. 

, 

U     00 

5.848 

8.771 

11.695 

17.  543 

6 

.003 

05 

.83!) 

.759 

.679 

.618 

7J 

.007 

07* 

.835 

.753 

.670 

.505 

10 

.012 

10 
15 

20 

.831 
.823 

5.815 

.746 

.662 

.493 

15 

.027 

8.722 

11.  629 

17. 444 

Latitude 
interval. 

Meridi- 
onal dis- 

2^ 
25 

.810 
.806 

.715    1        .621 
.709    1        .613 

.431 
.419 

30 

.798 

.  697    1        .  596 

.394 

1 

2 

Inch. 
1.620 
3.240 

35 

5.790 

8.685    i    11.580 

17.  370 

3 

4.861 

37J 

.786 

.678            .571 

.357 

4 

6.481 

40 

.782 

.672    1        ,563 

.345 

5 

8.101 

45 

.773 

.660 

.547 

.320 

6 

7 
8 

9.721 

11.  341 

12.  962 

50 

5.765 

8.647 

11.530 

17.  296 

9 

14.  582 

52* 

.761 

.641 

.523 

.284 

10 

16.  202 

55 

.757 

.635 

.614 

.271 

60 

.749 

.623 

.497 

.246 

Table  XSIV. — Area  of  quadrilaterals  of  Earth's  surface  of  1^  extent  in  latitude  and  longitude. 
[Prepared  by  E.  S.  AToodward.] 


Middle 
latitude 
of  <iuad- 
rilateral. 

Area 

in  square 

miles. 

Middle 
latitude 
of  quad- 
rilateral. 

Area 
in  square 

Middle 
latitude 
of  quad- 
rilateral. 

Area 

Middle 
latitude 
of  quad- 
rilateral. 

Area 
in  square 

Middle 
latitude 
of  quad- 
rilateral. 

Area 
in  square 

Middle 
latitude 
of  quad- 
rilateral. 

If! 

0 

00 

4752. 33 

15 

30 

4583.  92 

30 

30 

4109.  06 

45 

30 

3354.  01 

60     30 

2364.  34 

75 

30 

1205. 13 

0 

30 

52.16 

16 

00 

72.94 

31 

00 

4088.  21 

46 

00 

24.49 

61     00 

23.02 

76 

00 

1164.  49 

1 

00 

51.63 

16 

30 

6L61 

31 

30 

67.05 

46 

30 

3294.  71 

61    30 

2291.  51 

76 

30 

23.75 

1 

30 

50.75 

17 

00 

49.94 

32 

00 

45.57 

47 

00 

64.63 

62    00 

54.82 

77 

00 

1082.  91 

2 

00 

49.52 

17 

30 

37.93 

32 

30 

23.79 

47 

30 

34.39 

62    30 

17.94 

77 

30 

41.99 

2 

30 

47.93 

> 

3 

00 

46.00 

18 

00 

25.59 

33 

00 

01.69 

43 

00 

03.84 

63    00 

2180.  89 

78 

00 

1000. 99 

3 

30 

43.71 

18 

30 

12.90 

33 

30 

3979.  30 

48 

30 

3173.  04 

63    30 

43.66 

78 

30 

959.  90 

4 

00 

41.07 

19 

00 

4499.  87 

34 

00 

56.59 

49 

00 

41.99 

64    00 

06.26 

79 

00 

18.73 

4 

30 

38.08 

19 

30 

86.51 

34 

30 

33.59 

49 

30 

10.69 

64    30 

2063.  68 

79 

30 

877.  49 

5 

00 

34.74 

20 

00 

72.81 

35 

00 

10.28 

60 

00 

3079. 15 

65    00 

30.94 

80 

00 

36.18 

5 

30 

31.04 

20 

30 

58.78 

35 

30 

3886.  67 

50 

30 

47.37 

65    30 

1993.  04 

80 

30 

794.  79 

6 

00 

27.00 

21 

00 

44.41 

36 

00 

62.76 

51 

00 

15.34 

66    00 

54.97 

81 

00 

53.34 

6 

30 

22.61 

21 

30 

29.71 

36 

30 

38.56 

51 

30 

2983.  08 

66    30 

16.75 

31 

30 

11.83 

7 

00 

17.86 

22 

00 

14.67 

37 

00 

14.06 

52 

00 

50.  58 

67    00 

1878.37 

32 

00 

670.  27 

7 

30 

12.76 

22 

30 

4399.30 

37 

30 

3789.  26 

52 

30 

17.85 

67    30 

39.34 

82 

30 

28.64 

8 

00 

07.32 

23 

00 

83.60 

38 

00 

64.18 

63 

00 

2884.  88 

68    00 

1301. 16 

83 

00 

586.  97 

8 

30 

01.62 

23 

30 

67.57 

38 

30 

38.80 

53 

30 

51.68 

68    30 

1762.  33 

83 

30 

45.24 

9 

00 

4696. 38 

24 

00 

51.21 

39 

00 

13.14 

64 

00 

18.27 

69    00 

23.36 

84 

00 

03.  47 

9 

30 

88.89 

24 

30 

34.52 

39 

30 

3687. 18 

64 

30 

2734.  62 

69     30 

1684.  24 

84 

30 

461.  66 

10 

00 

82.05 

26 

00 

17.51 

40 

00 

60.95 

55 

00 

60.76 

70    00 

45.00 

85 

00 

19.81 

10 

30 

74.86 

25 

30 

00.17 

40 

30 

34.42 

65 

30 

16.67 

70    30 

05.62 

85 

30 

377.  93 

11 

00 

67.32 

26 

00 

4282.  50 

41 

00 

07.62 

56 

00 

2682.  37 

71    00 

1666. 10 

86 

00 

36.02 

11 

30 

59.43 

26 

30 

64.  51 

41 

30 

3580.  54 

56 

30 

47.85 

71    30 

26.46 

86 

30 

294.  08 

12 

00 

51.20 

27 

00 

46.20 

42 

00 

53.17 

57 

00 

13.13 

72    00 

1486.  70 

87 

00 

52.11 

12 

30 

42.63 

27 

30 

27.66 

42 

30 

25.54 

57 

30 

2578. 19 

72    30 

46.81 

37 

30 

10.12 

13 

00 

33.71 

28 

00 

08,61 

43 

00 

3497.  62 

58 

00 

43.05 

73    00 

06.31 

38 

00 

168. 12 

13 

30 

24.44 

28 

30 

4189.  33 

43 

30 

69.44 

68 

30 

07.70 

73     30 

1366.  69 

88 

30 

126. 10 

14 

00 

14.82 

29 

00 

69.74 

44 

00 

40.  98 

59 

00 

2472. 16 

74    00 

26.46 

39 

00 

84.07 

14 

30 

04.87 

29 

30 

49.83 

44 

30 

12.26 

59 

30 

36.42 

74     30 

1236. 12 

39 

30 

42.04 

15 

00 

4594.  57 

30 

00 

29.60 

45 

00 

3383.  27 

60 

00 

00.48 

75     00 

45.68 

90 

00 

00.00 

AREAS  OF  QUADEILATERALS. 


187 


Table  XXV. — Areas  of  quadrilaterals  of  Earth's  surface  of  30'  extent  in  latitude  and  longitude, 
[Prepared  by  E.  S.  "Woodward.] 


Middle 
latitude 

Area  in 

Middle 
latitude 

Area  in 

Middle 
latitude 

Area  in 

Middle 
latitude 

Area  in 

Middle 
latitude 

Area  in 

Middle 
latitude 

Area  in 

of  quad- 
rilateral. 

square 
miles. 

of  quad- 
rilateral. 

square 
mUes. 

of  quad- 
rilateral. 

square 
miles. 

of  quad- 
rilateral. 

square 
miles. 

of  quad- 
rilateral. 

square 
miles. 

of  quad- 
rilateral. 

miles. 

0 

30 

1188. 05 

30 

30 

1027.  27 

60 

30 

591. 09 

0 

15 

1188. 08 

30 

45 

1024.  68 

60 

45 

586. 50 

1 

00 

1187.  92 

31 

00 

1022.  06 

61 

00 

582. 01 

0 

45 

1188.  00 

31 

15 

1019. 43 

61 

15 

577. 45 

1 

30 

1187.  70 

3; 

30 

1016.  77 

61 

30 

572.  88 

1 

15 

1187.  82 

31 

45 

1014. 10 

61 

45 

568.  30 

2 

00 

1187.  39 

32 

00 

1011. 40 

62 

00 

563.  71 

1 

45 

1187.  56 

32 

15 

1008.  69 

62 

15 

559. 11 

3 

30 

1186.  99 

32 

30 

1005,96 

62 

30 

554.49 

2 
2 

15 
45 

1187.  20 
1186.  76 

32 

45 

1003. 20 

62 

45 

549.  86 

3 

00 

1186.  51 

33 

00 

1000.43 

63 

00 

545.23 

3 

15 

1186.  24 

33 

15 

997.64 

63 

15 

540. 58 

3 

30 

1185.  95 

33 

30 

994.  83 

63 

30 

635.  92 

3 

45 

1185.  62 

33 

46 

993.00 

63 

45 

531.  25 

i 

00 

1185.  28 

34 

00 

989. 16 

64 

00 

526.  57 

4 

15 

1184.  92 

34 

15 

986.  29 

64 

15 

621.  88 

4 

30 

1184.  53 

34 

30 

983.41 

64 

30 

517.17 

4 

45 

1184. 13 

34 

45 

980. 50 

64 

45 

512. 46 

5 

00 

1183.  70 

35 

00 

977.  58 

65 

00 

507.  74 

5 

15 

1183.24 

35 

15 

974.  64 

65 

16 

503.  01 

5 

30 

1182.77 

35 

30 

971.  68 

65 

30 

498.  26 

5 

45 

1182.  28 

35 

45 

968. 70 

65 

45 

493,  51 

6 

00 

1181.  76 

36 

00 

965.  70 

66 

00 

488.  75 

6 

15 

1181.  22 

36 

15 

962.  68 

66 

16 

483.  97 

6 

30 

1180.  66 

36 

30 

959.  65 

66 

30 

479. 19 

6 

45 

1180.  08 

36 

45 

956.  60 

66 

45 

474, 40 

7 

00 

1179.  48 

37 

00 

953.52 

67 

00 

469.  60 

7 

15 

1178. 85 

37 

15 

950. 43 

67 

15 

464.  78 

7 

30 

1178. 20 

37 

30 

947.32 

67 

30 

459.  96 

7 

45 

1177.  53 

37 

46 

944.  21 

67 

45 

455. 13 

8 

00 

1176.  84 

38 

00 

941.05 

68 

00 

450.  29 

8 

16 

1176. 13 

38 

15 

937.88 

67 

45 

455.13 

8 

30 

1175.  39 

38 

30 

984.71 

68 

30 

440.59 

8 

45 

1174.  63 

38 

45 

931.51 

68 

15 

445.45 

9 

00 

1173.  86 

39 

00 

928.29 

69 

00 

430.84 

9 

15 

1173.06 

39 

15 

935.  06 

68 

46 

435.  72 

9 

30 

1173. 23 

39 

30 

921.  SO 

69 

30 

421.  06 

9 

45 

1171. 39 

39 

45 

918. 53 

69 

15 

425.  96 

10 

00 

1170.52 

40 

00 

915.  25 

70 

00 

411.25 

10 

15 

1169.  63 

40 

15 

911.  94 

69 

45 

416.16 

10 

30 

^ 1168.  73 

40 

30 

908.  61 

70 

30 

401. 41 

10 

45 

1167.  80 

40 

45 

905. 27 

70 

16 

406.  34 

11 

00 

1166.  84 

41 

00 

901.91 

71 

00 

391.  53 

11 

15 

1165.  86 

41 

15 

898.  64 

70 

45 

396.  47 

11 

30 

1164.  86 

41 

30 

895. 14 

71 

30 

381.  62 

11 

45 

1163.  85 

41 

45 

891. 73 

71 

15 

386.  58 

12 

00 

1162.  81 

42 

00 

888.  30 

72 

00 

371.  68 

12 

15 

1161.  75 

42 

15 

884.85 

71 

45 

376,  65 

12 

30 

1160.67 

42 

30 

881.  39 

72 

30 

361. 71 

12 

45 

1159. 56 

42 

45 

877.  91 

72 

15 

366,  70 

13 

00 

1158.  44 

43 

00 

874.  41 

73 

00 

351.  71 

13 

15 

1157.  29 

43 

15 

870.  90 

72 

45 

356.71 

13 

30 

1156. 12 

43 

30 

867.  37 

73 

30 

341.  68 

13 

45 

1154.  93 

43 

45 

863.  82 

73 

16 

346.  69 

14 

00 

1153.72 

44 

00 

860.  25 

74 

00 

331.62 

14 

15 

1152. 48 

44 

15 

856.  67 

73 

45 

336. 65 

14 

30 

1151.  23 

44 

30 

853.07 

74 

30 

321.  53 

14 

45 

1149,  95 

44 

45 

849, 46 

74 

16 

326,  58 

15 

00 

1148.  65 

45 

00 

845.82 

75 

00 

311.42 

15 

15 

1147.  33 

45 

15 

842. 18 

74 

45 

316,48 

15 

30 

1145.99 

45 

30 

838.  51 

75 

30 

301.28 

15 

45 

1144. 63 

45 

45 

834.  83 

75 

15 

306,  36 

16 

00 

1143.  25 

46 

00 

831. 13 

76 

00 

291. 12 

16 

15 

1141.  84 

46 

15 

827.42 

75 

45 

296,21 

16 

30 

1140. 41 

46 

30 

823.  68 

76 

30 

280. 94 

16 

45 

1138. 96 

46 

45 

819.  94 

76 

15 

286,04 

17 

00 

1137.  50 

47 

00 

816. 18 

77 

00 

270.73 

17 

15 

1136.  00 

47 

15 

812. 40 

76 

46 

275, 84 

17 

30 

U34.49 

47 

30 

808.  60 

77 

30 

260.  50 

17 

45 

1132.  96 

47 

45 

804.  79 

77 

15 

265,  62 

18 

00 

1131.  41 

48 

00 

800.  97 

78 

00 

250.  25 

18 

15 

1120.  83 

48 

15 

797. 13 

77 

45 

255.  38 

18 

30 

1128.24 

48 

30 

793.  27 

78 

30 

239.  98 

18 

45 

1126.  62 

48 

45 

789.  39 

78 

15 

215.12 

19 

00 

1124.  98 

49 

00 

785.  50 

79 

00 

229.  68 

19 

15 

1123.32 

49 

15 

781.  60 

78 

45 

234.83 

19 

30 

1121.64 

49 

30 

777.  68 

79 

30 

219.  37 

19 

45 

1119.  93 

49 

45 

773.  74 

79 

15 

224.53 

20 

00 

1118.21 

50 

00 

769.  79 

80 

00 

209.  05 

20 

15 

1116.  47 

50 

15 

765.83 

79 

45 

214.  21 

20 

30 

1114. 71 

50 

30 

761.  85 

80 

30 

198.  70 

20 

45 

1112.  92 

50 

45 

757.85 

80 

15 

203. 88 

21 

00 

1111.11 

51 

00 

753. 84 

81 

00 

188.34 

21 

15 

1109,  28 

51 

15 

749.  82 

80 

45 

193.  52 

21 

30 

1107.44 

51 

30 

745.78 

81 

30 

177.  96 

21 

45 

1105.  57 

51 

45 

741.72 

81 

15 

183. 15 

22 

00 

1103.  68 

62 

00 

737.  65 

82 

00 

167. 57 

22 

15 

1101.  77 

52 

15 

733.  67 

81 

45 

172.  77 

22 

30 

1099.84 

52 

30 

729.  47 

82 

30 

157. 16 

22 

45 

1097.  88 

52 

45 

725.  36 

82 

15 

162.  37 

23 

00 

1095. 91 

53 

00 

721.  23 

83 

00 

146.  74 

23 

15 

1093.  93 

53 

15 

717.  08 

82 

46 

151.95 

23 

30 

1091.90 

53 

30 

712.  93 

83 

30 

136.  31 

23 

45 

1089.  87 

53 

45 

708.  76 

S3 

15 

141.  53 

24 

00 

1087.  81 

54 

00 

704.  57 

84 

00 

125.  87 

24 

15 

1085.  74 

54 

15 

700. 38 

83 

45 

131.  09 

24 

30 

1083. 64 

54 

30 

696. 16 

84 

30 

115.  42 

24 

45 

1081.  52 

54 

45 

691.  94 

84 

15 

120.  64 

25 

00 

1079.  39 

55 

00 

687.  70 

85 

00 

104.  95 

25 

15 

1077.  23 

55 

16 

683.44 

84 

45 

110. 18 

25 

30 

1075.  05 

55 

30 

679. 17 

85 

30 

94.48 

25 

45 

1072.  85 

55 

45 

674.  89 

86 

15 

99.72 

26 

00 

1070.  64 

56 

00 

670.  60 

86 

00 

84.01 

26 

15 

1068.  40 

56 

15 

666.  29 

85 

46 

89.35 

26 

30 

1066. 14 

56 

30 

661.97 

86 

30 

73.52 

26 

45 

1063.  86 

56 

45 

657.64 

86 

15 

78.76 

27 

00 

1061.  56 

57 

00 

653. 29 

87 

TO 

63.03 

27 

15 

1059.  24 

57 

15 

648.93 

86 

45 

68.37 

27 

30 

1056.  90 

57 

30 

644.55 

87 

30 

52.53 

27 

45 

1054. 54 

57 

45 

640. 17 

87 

15 

57.78 

28 

00 

1052. 16 

58 

00 

635.  77 

88 

00 

42.03 

28 

15 

1049.  76 

58 

15 

631.  36 

87 

45 

47.28 

28 

30 

1047.34 

58 

30 

626.  93 

88 

30 

31.53 

28 

45 

1044.  90 

58 

45 

622.49 

88 

15 

36.78 

29 

00 

1U42.  44 

59 

00 

618.  05 

89 

00 

21.02 

29 

15 

1039.  97 

59 

15 

613.  59 

88 

45 

26.27 

29 

30 

1037.  47 

59 

30 

609. 11 

89 

30 

10.51 

29 

45 

1034.  95 

59 

45 

604.  62 

89 

15 

16.76 

30 

00 

1032.  41 

60 

00 

600. 13 

90 

00 

00.00 

30 

15 

1039.  85 

60 

15 

595.  62 

89 

45 

5.26 

188 


A  MANUAL  OF  TOPOGEAPHIG  METHODS. 


Table  XXVI. — Areas  of  quadrilaterals  of  Earth's  surface  of  16'  extent  in  latitme  and  longitude. 
[Prepared  by  K.  S.  "Woodward.] 


Middle 
latitude 
of  quadri- 
lateral. 

A.rea  in 

Middle 
latitude 

Area  in 

MidcUe 
latitude 

Area  in 

Middle 
latitude 

Area  in 

Middle 
latitude 

\roa  in 

Middle 
latitude 

Area  in 

square 
miles. 

of  quadri- 
lateral. 

square 
miles. 

of  quadri- 
lateral. 

square 
mues. 

of  quadri- 
lateral. 

square 
miles. 

of  quadri- 
lateral. 

square 
miles. 

of  quadri- 
lateral. 

square 
miles. 

0  07  30 

297.02 

8  15  00 

294. 03 

16  22  30 

285. 28 

24  30  00 

270. 91 

32  37  30 

251. 15 

40  45  00  . 

226.  32 

0  15  00 

297.02 

8  22  30 

293. 94 

16  30  00 

285. 10 

24  37  30 

270.  65 

32  45  00 

250.80 

40  52  30 

225.  90 

0  22  30 

297.  02 

8  30  00 

293.85 

16  37  30 

284.  92 

24  45  00 

270.  38 

32  52  30 

250.  45 

41  00  00 

225.  48 

0  30  00 

297.  01 

8  37  30 

293.  75 

16  45  00 

284.  74 

24  53  30 

270.11 

33  00  00 

250. 11 

41  07  30 

225.  06 

0  37  30 

297.01 

8  45  00 

293.  66 

16  52  30 

284.  56 

25  00  00 , 

269.  85 

33  07  30 

249.  76 

41  15  00 

224.64 

0  45  00 

297.  00 

8  52  30 

293.  56 

17  00  00 

284.  38 

25  07  30 

269.  58 

33  15  00 

249.41 

41  22  30 

224.  21 

0  52  30 

296. 99 

9  00  00 

293.47 

17  07  30 

284. 19 

25  15  00 

269.  31 

33  22  30 

249.  06 

41  30  00 

223.  79 

1  00  00 

296.  98 

9  07  30 

293.  37 

17  15  00 

284.  00 

25  22  30 

269.  04 

33  30  00 

248.  71 

41  37  30 

223.  36 

1  07  30 

290.  97 

9  15  00 

293.  27 

17  22  30 

283.  81 

25  30  00 

268.  76 

33  37  30 

248.  36 

41  45  00 

222.  93 

1  15  00 

296.  96 

9  22  30 

293. 16 

17  30  00 

283.  62 

25  37  30 

268.  49 

33  45  00 

248.  00 

41  52  30 

222.  50 

1  22  30 

296.  94 

9  30  00 

293.  06 

17  37  30 

283.  43 

25  45  00 

268.  21 

33  52  30 

247.  65 

42  00  00 

222.  08 

1  30  00 

296.  93 

9  37  30 

292.95 

17  45  00 

283.  24 

25  52  30 

267.  94 

34  00  00 

247.  29 

42  07  30 

221.  65 

1  37  30 

296. 91 

9  45  00 

292.85 

17  52  30 

2S3.  05 

26  00  00 

267.  66 

34  07  30 

246.  93 

42  15  00 

221.  21 

1  45  00 

296.  89 

9  52  30 

292.74 

18  00  00 

282.  86 

26  07  30 

267.  38 

34  15  00 

246.  57 

42  22  30 

220.  78 

1  52  30 

296.87 

10  00  00 

292.  63 

18  07  30 

282.  66 

20  15  00 

267. 10 

34  22  30 

246.  21 

42  30  00 

220.  35 

2  00  00 

296.  85 

10  07  30 

292.  52 

18  15  00 

282.  46 

26  22  30 

266.82 

34  30  00 

245.  85 

42  37  30 

219.  91 

2  07  30 

296. 82 

10  15  00 

292.41 

18  22  30 

282.  26 

26  30  00 

266.  54 

34  37  30 

245.  49 

42  45  00 

219.  48 

2  15  00 

296.  80 

10  22  30 

292.  30 

18  30  00 

282.  06 

26  37  30 

266.  25 

34  45  00 

245. 13 

42  52  30 

219.  04 

2  22  30 

296.  77 

10  30  00 

292. 19 

18  37  30 

281.  86 

26  45  00 

265.  97 

34  -52  30 

244.76 

43  00  00 

218.  60 

2  30  00 

296.  75 

10  37  30 

292.  07 

18  45  00 

281.66 

26  52  30 

265.  68 

35  00  00 

244.  40 

43  07  30 

218. 10 

2  37  30 

296.  72 

10  45  00 

291.  95 

18  52  30 

281.45 

27  00  00  '  265.  39 

35  07  30 

244.03 

43  15  00 

217.  73 

2  45  00 

296.  69 

10  52  30 

291.  83 

19  00  00 

281.  25 

27  07  30 

265. 10 

35  15  00 

243.  06 

43  22  30 

217.  28 

2  52  30 

296.  66 

11  00  00 

291.  71 

19  07  30 

281.  04 

27  15  00 

264.  81 

35  22  30 

243.  29 

43  30  00 

216.84 

3  00  00 

296.  63 

11  07  30 

291.59 

19  15  00 

280.  83 

27  22  30 

264.52 

35  30  00 

242.  92 

43  37  30 

216.  40 

3  07  30 

296.  60 

U  15  00 

291.47 

19  22  30 

280.  62 

27  30  00 

264. 23 

35  37  30 

242.55 

43  45  00 

215.  96 

3  15  00 

296.  56 

11  22  30 

291.  34 

19  30  00 

280.  41 

27  37  30 

263.  93 

35  45  00 

242.18 

43  52  30 

215.  51 

3  22  30 

296.  53 

11  30  00 

291.  22 

19  37  30 

280.  20 

27  45  00 

263.  64 

35  52  30 

241.  80 

44  00  00 

215.  06 

3  30  00 

296.49 

11  37  30 

291.  09 

19  45  00 

279. 99 

27  52  30 

263.  34 

36  00  00 

241.43 

44  07  30 

214.  61 

3  37  30 

296.45 

11  45  00 

290.  96 

19  52  30 

279.  77 

28  00  00 

263.  04 

36  07  30 

241.  05 

44  15  00 

214. 17 

3  45  00 

296,41 

11  52  30 

290.  83 

20  00  00 

279.  55 

28  07  30 

262.  74 

36  15  00 

240.  67 

44  22  30 

213.  72 

3  52  30 

296.  36 

12  00  00 

290.  70 

20  07  30 

279.  34 

28  15  00  '  262.  44 

36  22  30 

240.  29 

44  30  00 

213.  27 

4  00  00 

296.  32 

12  07  30 

290.57 

20  15  00 

279. 12 

28  22  30   262.  14 

36  30  00 

239.  91 

44  37  30 

212.  82 

4  07  30  296.  28 

12  15  00 

290.44 

20  22  30 

278.  OQ 

28  30  00  1  261.  84 

36  37  30 

239.  53 

44  45  00 

212.  37 

4  15  00 

296.  23 

12  22  30 

290.  30 

20  30  00 

278.  68 

28  37  30 

261.  53 

36  45  00 

239. 15 

44  52  30 

211.91 

4  22  30 

296. 18 

12  30  00 

290. 17 

20  37  30 

278.  46 

28  45  00 

261.  23 

36  52  30 

238.  77 

45  00  00 

211.  46 

4  30  00 

296. 13 

12  37  30 

290.  03 

20  45  00 

278.  23 

28  52  30 

260.  92 

37  00  00 

238.  38 

45  07  30 

211.00 

4  37  30 

296.  08 

12  45  00 

289.  89 

20  52  30 

278.00 

29  00  00 

260.  01 

37  07  30 

237.  99 

45  15  00 

210.  55 

4  45  00 

296.  03 

12  52  30 

289.  75 

21  00  00 

277.  78 

29  07  30 

260.  30 

37  15  00 

237.  61 

45  22  30 

210.  09 

4  52  30 

295. 98 

13  00  00 

289.  61 

21  07  30 

277.  55 

29  15  00 

259.  99 

37  22  30 

237.  22 

45  30  00 

209.  63 

5  00  00 

295.  93 

13  07  30 

289.47 

21  15  00 

277.  32 

29  22  30 

259.  68 

37  30  00 

236.  83 

45  37  30 

209. 17 

5  07  30 

295.  87 

13  15  00 

289.  33 

21  22  30 

277.  09 

29  3i)  00 

259.  37 

37  37  30 

236.  44 

45  45  00 

208.  71 

5  15  00 

295.  81 

13  22  30 

289. 18 

21  30  00 

276.  86 

29  37  30 

259.  05 

37  45  00 

236.  05 

45  52  30 

208.  25 

5  22  30 

295.75 

13  30  00 

289.  03 

21  37  30 

276.  63 

29  45  00 

258.  74 

37  52  30 

235.  60 

46  00  00 

207.  78 

5  30  00 

295.  69 

13  37  30 

288.  88 

21  45  00 

276.  39 

29  52  30 

258.  42 

38  00  00 

235.  26 

46  07  30 

207.  32 

5  37  30 

295.  63 

13  45  00 

288.  73 

21  52  30 

276. 16 

30  00  00 

258. 10 

38  07  30 

234.  87 

46  15  00 

206.  86 

5  45  00 

295.  57 

13  52  30 

288. 58 

22  00  00 

275.  92 

30  07  30 

257.  78 

38  15  00 

234.47' 

46  22  30 

206.  39 

5  52  30 

295.  51 

14  00  00 

288.  43 

22  07  30 

275.  68 

30  15  OO 

257.  46 

38  22  30 

234.  07 

46  30  00 

205.  92 

6  00  00 

295.44 

14  07  30 

288.  28 

22  15  00 

275.44 

30  22  30 

257. 14 

38  30  00 

233.  68 

46  37  30 

205. 45 

6  07  30 

295.  37 

14  15  00 

288. 12 

22  22  30 

275.  20 

30  30  00 

256.  82 

38  37  30 

233.  28 

46  45  00 

204.  99 

6  15  00 

295.  31 

14  22  30 

287.96 

22  30  00 

274.  96 

30  37  30 

256.  49 

38  45  00 

232.  88 

46  52  30 

204.  52 

6  22  30 

295.24 

14  32  CO 

287.  81 

22  37  30 

274.  72 

30  45  00 

256. 17 

38  52  30 

232. 48 

47  00  00 

204.  05 

6  30  00 

295. 17 

14  37  30 

287.  65 

22  45  00 

247.47 

30  52  30 

255.  84 

39  00  00  ;  232.  07 

47  07  30 

203.  57 

6  37  30 

295.  09 

14  45  00 

287.  49 

22  52  30 

274.  22 

31  00  00 

255.  52 

39  07  30 

231.  67 

47  15  00 

203. 10 

6  45  00 

295.  02 

14  52  30 

287. 33 

23  00  00 

273.  98 

31  07  30 

255. 19 

39  15  00 

231.  27 

47  22  30 

202.  63 

6  52  30 

294.  95 

15  00  00 

287. 17 

23  07  30 

273.  73 

31  15  00 

254. 86 

39  22  30 

230.  86 

47  30  00 

202. 15 

7  00  00 

294.  87 

15  07  30 

287. 00 

23  15  00 

273.48 

31  22  30 

254.  53 

39  30  00 

230.45 

47  37  30 

201.  67 

7  07  30 

294.  79 

15  15  00 

286.  83 

23  22  30 

273.  23 

31  30  00 

254. 19 

39  37  30 

230.  04 

47  45  00 

201.  20 

7  15  00 

294.  71 

15  22  30 

286.  67 

23  30  00 

2T2.  98 

31  37  30 

253.  86 

39  45  00 

229.  03 

47  52  30 

200.  72 

7  22  30 

294.  63 

15  30  00 

286.  50 

23  37  30 

272.  72 

31  45  00 

253.  53 

39  52  30 

229.  22 

48  00  00 

200.  24 

7  30  00 

294.  55 

15  37  30 

286. 33 

23  45  00 

272.47 

31  52  30 

253. 19 

40  00  00 

228.  81 

48  07  30 

199.  76 

7  37  30 

294.  47 

15  45  00 

286. 16 

23  52  30 

272.  21 

32  00  00 

252.  85 

40  07  30 

228.  40 

48  15  00 

199.  28 

7  45  00 

294.  39 

15  52  30 

285. 99 

24  00  00 

271.  95 

32  07  30 

252.  51 

40  15  00 

227.  99 

48  22  30 

198.80 

7  52  30 

294.  30 

10  00  00 

285.  82 

24  07  30 

271.69 

32  15  00 

252. 17 

40  22  30 

227.  57 

48  30  00 

198.  32 

8  00  00 

294. 21 

16  07  30 

285.  64 

24  15  00 

271.  44 

32  22  30 

251.  83 

40  30  00 

227. 15 

48  37  30 

197.  83 

8  07  30 

294. 12 

16  15  00 

285.  46 

24  22  30 

271. 17 

32  30  00 

251.  49 

40  37  30 

226.  73 

48  45  00 

197.  35 

AEEAS  OF  QUADRILATERALS. 


189 


Table  XSVI. Areas  of  qitadrilaterals  of  Earth's  surface  of  15'  extent  in  latitude  and  longitude — Cont'd. 

[Prepared  by  E.  S.  "Woodward.] 


Middle 
latitude 
of  quadri- 
lateral. 

ireain 
square) 
milea. 

Middle 
latitude 
of  quadri- 
lateral. 

A.rea  in 
square 
miles. 

MidiUe 
latitude 
of  quadri- 
lateral. 

Alreaiu 
square 
miles. 

Middle 
Latitude 
of  quadri- 
lateral. 

Arcaiu 
square 
miles. 

Middle 

latitudi' 

of  ciuadri- 

latfral. 

A.rea  in 
square 
miles. 

Middle 
latitude 
of  quadri- 
lateral. 

Area  in 
square 
miles. 

35.38 

48  52  30 

196.  86 

55  45  00 

168. 72 

62  37  30 

138.04 

69  30  00 

105.  27 

76  22  30 

70.87 

83  15  00 

49  00  00 

196.  38 

55  52  30 

153. 19 

62  45  00 

137. 47 

69  37  30 

104.  65 

76  30  00 

70.24 

83  23  30 

34.73 

49  07  30 

195.  89 

56  00  00 

167.  65 

63  52  30 

136.  89 

69  45  00 

104.  04 

76  37  30 

69.60 

83  30  00 

34.08 

49  15  00 

195.  40 

56  07  30 

167. 11 

63  on  00 

136.  31 

69  52  30 

103.  43 

76  46  00 

68.96 

83  37  30 

33.42 

49  22  30 

194.  91 

56  15  00 

166.  57 

03  07  30 

136.  73 

70  00  00 

102.  81 

76  52  30 

68.32 

83  45  00 

32.77 

49  30  00 

194.  42 

56  23  30 

106.  03 

63  15  00 

135. 15 

70  07  30 

102.  20 

77  00  00 

67.68 

83  52  30 

33.12 

49  37  30 

193.  93 

56  30  00 

165.  49 

63  22  30 

134.  66 

70  15  00 

101.  59 

77  07  30 

67.04 

84  00  00 

31.47 

49  45  00 

193.  44 

56  37  30 

164.95 

63  30  00 

133.  98 

70  22  30 

100.  97 

77  15  00 

66.41 

84  07  30 

30.81 

49  52  30 

192.  94 

56  45  00 

164.  41 

63  37  30 

133.  40 

70  30  00 

100.  35 

77  22  30 

65.77 

84  15  00 

30.16 

50  00  00 

192.  45 

56  52  30 

163.  87 

63  45  00 

132.  81 

70  37  30 

99.74 

77  30  00 

65.13 

84  23  30 

29.51 

50  07  30 

191.95 

57  00  00 

163.  32 

63  52  30 

132.33 

70  45  00 

99.13 

77  37  30 

64.49 

84  30  00 

28.86 

50  15  00 

191.46 

57  07  30 

162.  78 

64  00  00 

131.  64 

70  52  30 

98.50 

77  45  00 

63.86 

84  37  30 

28.20 

50  22  30 

190.  96 

57  15  00 

162.  23 

64  07  30 

131.  06 

71  00  00 

97.88 

77  52  30 

63.20 

84  46  00 

37.54 

50  30  00 

190.  46 

57  22  30 

161.  68 

64  15-00 

130.  47 

71  07  30 

97.36 

78  00  00 

62.56 

84  52  30 

36.89 

60  37  30 

1S9.  96 

57  30  00 

161. 14 

64  22  30 

129.  88 

71  15  00 

96.65 

78  07  30 

61.92 

85  00  00 

26.24 

50  45  00 

189.  46 

57  37  30 

160.  59 

64  30  00 

129.  29 

71  22  30 

96.03 

78  15  00 

61.28 

85  07  30 

25.58 

50  52  30 

188.  nii 

57  45  00 

160.  04 

64  37  30 

128. 70 

71  30  00 

95.41 

78  22  30 

60.64 

85  15  00 

24.93 

51  00  00 

188.40 

■  57  52  30 

159.49 

64  45  00 

128. 12 

71  37  30 

94.78 

78  30  00 

60.00 

85  22  30 

34.37 

51  07  30 

187.  96 

58  00  00 

158.  94 

64  52  30 

137.  53 

71  46  00 

94.16 

78  37  30 

59.35 

85  30  00 

23.62 

51  15  00 

187.  46 

58  07  30 

158.  39 

65  00  00 

126.  94 

71  52  30 

93.54 

78  45  00 

68.71 

85  37  30 

22.97 

51  22  30 

186.  95 

58  15  00 

157.  84 

65  07  30 

126.  34 

72  00  00 

92.92 

78  62  30 

58.06 

85  45  00 

22.31 

51  30  00 

186.  45 

58  22  30 

157.  29 

65  16  00 

125.  75 

72  07  30 

93.30 

79  00  00 

57.43 

85  52  30 

21.66 

51  37  30 

185.  94 

58  30  00 

156.  73 

65  23  30 

125. 16 

72  15  00 

91.68 

79  07  30 

56.78 

86  00  00 

21.00 

51  45  00 

185,  43 

58  37  30 

156. 18 

65  30  00 

124.  57 

72  32  30 

91.05 

•79  15  00 

56.13 

86  07  30 

20.35 

51  52  30 

184.  92 

58  45  00 

155.  63 

65  37  30 

123.  97 

72  30  00 

90.43 

79  22  20 

55.49 

86  15  00 

19.69 

52  00  00 

184.  41 

58  52  30 

155.  07 

65  45  00 

123.  38 

72  37  30 

89.80 

79  30  00 

54.84 

86  22  30 

19.04 

52  07  30 

183.  90 

59  00  00 

154.  51 

65  52  30 

123,  78 

73  46  00 

89.18 

79  37  30 

54.20 

86  30  00 

18.38 

52  15  00 

183.  39 

69  07  30 

153.  96 

66  00  00 

122. 19 

72  52  30 

88.55 

79  45  00 

63.55 

86  37  30 

17.73 

52  22  30 

182  88 

59  15  00 

153.  41 

66  07  30 

121.  59 

73  00  00 

87.93 

79  52  30 

52.91 

86  45  00 

17.07 

52  30  00 

182.37 

59  22  30 

152.  84 

66  15  00 

120. 99 

73  07  30 

87.30 

80  00  00 

52.26 

86  52  30 

16.41 

52  37  30 

181.  85 

59  30  00 

152.  28 

66  22  30 

120.  40 

73  16  00 

86.67 

80  07  30 

51.63 

87  00  00 

15.76 

52  45  00 

181.  34 

59  37  30 

151.  72 

66  30  00 

U9.80 

73  22  30 

86.05 

80  15  00 

50.97 

87  07  30 

16.10 

52  52  30 

180.  S2 

59  45  00 

151. 16 

66  37  30 

119.  20 

'73  30  00 

85.43 

80  22  30 

50.32 

87  15  00 

14.44 

53  00  00 

180.  31 

59  52  30 

150.  60 

66  45  00 

118.  60 

73  37  30 

84.79 

80  30  00 

49.68 

87  22  30 

13.79 

53  07  30 

179.  79 

60  00  00 

150.  03 

66  52  30 

118.  00 

73  45  00 

84.16 

80  37  30 

49.03 

87  30  00 

13.13 

53  15  00 

179.  27 

60  07  30 

149.  47 

67  00  00 

117.  40 

73  52  30 

83.53 

80  46  00 

48.38 

87  37  30 

12.48 

53  22  30 

178.  75 

60  15  00 

148.  91 

67  07  30 

116.80 

74  00  00 

82.91 

80  52  30 

47.73 

87  45  00 

11.82 

53  30  00 

178.  23 

60  23  30 

148.  34 

67  15  00 

116.20 

74  07  30 

83.38 

81  00  00 

47.08 

87  52  30 

11.16 

53  37  30 

177.71 

60  30  00 

147.  77 

67  22  30 

115.  59 

74  16  00 

81.65 

81  07  30 

46.44 

88  00  00 

10.51 

53  45  00 

177. 19 

60  37  30 

147.  21 

67  30  00 

114.  99 

74  22  30 

81.01 

81  15  00 

45.79 

88  07  30 

9.85 

53  52  30 

176.  67 

60  45  00 

146.  64 

67  37  30 

114.  39 

74  30  00 

80.38 

81  22  30 

45.14 

88  15  00 

9.20 

54  00  00 

176. 14 

60  52  30 

146.  07 

67  45  00 

113.  78 

74  37  30 

79.75 

81  30  00 

44.49 

88  22  30 

8.54 

54  07  30 

175.  62 

61  00  00 

145.  50 

67  52  30 

113.18 

74  45  00 

79.12 

81  37  30 

43.84 

88  30  00 

7.88 

54  15  00 

175. 10 

61  07  30 

144.93 

68  00  00 

112.57 

74  53  30 

78.49 

81  45  00 

43.19 

88  37  30 

7.22 

54  22  30 

174.  57 

61  15  00 

144.  36 

68  07  30 

111.  97 

75  00  00 

77.86 

81  53  30 

42.64 

88  45  00 

6.57 

54  30  00 

174.  04 

61  22  30 

143.  79 

68  15  00 

111.36 

76  07  30 

77.22 

82  00  00 

41.89 

88  53  30 

5.91 

54  37  30 

173.  51 

61  30  00 

143.  22 

68  22  30 

110.  76 

75  16  00 

76.59 

82  07  30 

41.24 

89  00  00 

6.36 

54  45  00 

172.  99 

'     61  37  30 

142.  65 

68  30  00 

110. 15 

76  22  30 

75.95 

82  15  00 

40.59 

89  07  30 

4.60 

54  52  30 

172.  46 

61  47  00 

142.  08 

63  37  30 

109.  54 

75  30  00 

75.33 

82  22  30 

39.94 

89  15  00 

3.94 

55  00  00 

171.  93 

61  52  30 

141. 50 

68  45  00 

108.  93 

75  37  30 

74.69 

83  30  00 

39.29 

89  22  30 

8.28 

55  07  30 

171.  39 

i     62  00  00 

140.  93 

68  52  30 

108.  32 

75  45  00 

74.05 

83  37  30 

38.64 

89  30  00 

2.63 

55  15  00 

170.  86 

63  07  30 

140.  35 

69  00  00 

107.  71 

75  52  30 

73.42 

83  45  00 

37.99 

89  37  30 

1.97 

55  22  30 

170.  33 

62  15  00 

139.  78 

69  07  30 

107. 10 

76  00  00 

72.78 

82  52  30 

37.34 

89  45  00 

1.31 

55  30  00 

169.  79 

1     62  22  30 

139.  20 

69  15  00 

106.  49 

76  07  30 

72.14 

83  00  00 

36.69 

89  52  30 

0.66 

55  37  30 

169.  26 

1     62  30  00 

138.  62 

69  22  30 

105.  88 

76  15  00 

71.61 

83  07  30 

36.03 

190         A  MANUAL  OF  TOPOGEAPHIC  METHODS. 

TA.BLE  XXVII.— fitf/ofs /or  the  compuiation  of  (jeodetic  latitudes,  longittides,  and  azimuths. 

[From  Appendix  No.  7,  Kcport  U.  S.  Crast  and  Geodetic  Survey,  1884.] 

LATITUDE  25^. 


log.  A 


log.  B 


Iditt.  1"  =  — 0.06  diff.l"  =  — 0.16 


log.C 
diff.l"  =  +  0.54 


log.  D  log.  E 

diff.  1"  =  +0.03  diff.  1"  =  +  0.04 


FACTOES  FOE  COMPUTATION  OF  GEODETIC  POSITIONS.       191 


Table  XXVXI. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  26°. 


log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

Latitude. 

(liflF.l"  =  -0.06 

diff.  1"— — 0.17 

diff.  1"= +0.53 

diff.  l"  =  +0.03 

diff.  1"= +0.04 

26  00 

8.  509  4439 

8. 511  8283 

1. 09400 

2.2885 

5.  8458 

1 

36 

72 

432 

87 

61 

33 

62 

464 

89 

63 

3 

29 

52 

496 

91 

66 

4 

26 

42 

527 

93 

69 

05 

22 

32 

559 

95 

71 

6 

19 

32 

691 

97 

74 

7 

IG 

12 

633 

99 

77 

8 

12 

01 

655 

2.  2901 

79 

9 

09 

8.  5U  8191 

687 

03 

82 

10 

8.  509  4406 

8.511  818! 

1.09718 

2. 2905 

6.8485 

11 

02 

71 

750 

07 

88 

12 

8.  509  4399 

61 

782 

09 

90 

la 

95 

51 

814 

11 

93 

14 

92 

40 

845 

13 

96 

15 

88 

30 

877 

15 

98 

16 

85 

20 

909 

17 

5.  8501 

17 

82 

10 

940 

19 

04 

18 

78 

00 

972 

20 

06 

19 

8.  511  8089 

1. 10004 

22 

09 

20 

8.  509  4372 

8.  511  8079 

1. 10036 

2. 2924 

6.  8512 

21 

08 

69 

067 

26 

14 

22 

65 

59 

099 

28 

17 

23 

61 

48 

130 

30 

20 

24 

58 

38 

162 

32 

22 

25 

54 

28 

194 

34 

25 

26 

51 

18 

225 

36 

28 

27 

48 

08 

257 

38 

30 

28 

44 

8. 611  7997 

288 

40 

33 

29 

41 

87 

820 

42 

36 

30 

8.  609  4337 

8. 611  7977 

1. 10351 

2.2944 

5. 8539 

31 

34 

67 

383 

46 

41 

32 

31 

56 

•   414 

47 

44 

33 

27 

46 

446 

49 

47 

34 

24 

36 

477 

51 

49 

35 

20 

25 

609 

53 

52 

36 

17 

15 

540 

55 

55 

37 

13 

05 

571 

57 

57 

38 

10 

8.  511  7895 

603 

69 

60 

39 

07 

84 

634 

61 

63 

40 

8.  509  4303 

8.  511  7874 

1. 10666 

2. 2963 

5.  8566 

41 

00 

64 

697 

65 

68 

42 

8. 509  4296 

53 

728 

66 

71 

43 

93 

43 

760 

68 

74 

44 

89 

33 

791 

70 

76 

45 

86 

22 

832 

72 

79 

46 

83 

12 

854 

74 

82 

47 

79 

02 

885 

76 

85 

48 

76 

8. 511  7791 

9J6 

78 

87 

49 

73 

81 

947 

80 

90 

50 

8. 609  4269 

8. 511  7771 

1. 10979 

2.3981 

5.  8593 

51 

05 

60 

1. 11010 

83 

95 

52 

62 

50 

041 

85 

98 

53 

5S 

40 

073 

87 

5.8601 

54 

66 

29 

103 

89 

04 

55 

52 

19 

134 

91 

06 

56 

48 

09 

166 

93 

09 

57 

46 

8.511  7698 

197 

94 

12 

58 

41 

88 

228 

96 

14 

59 

38 

77 

259 

98 

17 

60 

8.  509  4234 

8. 511  7667 

1. 11290 

2.3000 

5. 8630 

192 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  computation  of  geodetic  lalituclcs,  longitudes,  and  azimuths — Coutinued, 

LATITUDE  27°. 


log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

latitude. 

difl'.  1"=— 0.06 

diir.l"=-0.18 

diff.  1"=+0.51 

diff.l"=+0.03 

diff.  1"= +0.05 

27  00 

8. 509  4234 

8.511  7667 

1. 11290 

2.  3000 

5. 8620 

1 

31 

57 

321 

02 

23 

2 

27 

46 

352 

04 

25 

3 

24 

36 

383 

06 

28 

4 

20 

25 

414 

07 

31 

5 

17 

15 

445 

09 

34 

6 

13 

05 

476 

11 

36 

7 

10 

8. 511  7594 

507 

13 

39 

8 

06 

84 

538 

15 

42 

9 

03 

73 

569 

17 

44 

10 

8. 509  4200 

8.  511  7563 

1. 11600 

2.3018 

5.  8647 

11 

8.  509  4196 

53 

631 

20 

50 

12 

93 

42 

662 

22 

53 

13 

89 

32 

693 

24 

55 

14 

86 

21 

724 

26 

58 

15 

82 

11 

755 

27 

01 

36 

79 

00 

786 

29 

64 

17 

75 

8.  511  7490 

817 

31 

66 

18 

79 

848 

33 

69 

19 

68 

69 

878 

35 

72 

20 

8.509  4165 

8. 511  7458 

1. 11909 

2.  3037 

5.  8675 

21 

61 

48 

940 

38 

77 

22 

58 

37 

971 

40 

80 

23 

54 

27 

1. 12002 

42 

83 

24 

51 

16 

032 

44 

86 

25 

47 

06 

063 

45 

88 

26 

44 

8.  511  7395 

094 

47 

91 

27 

40 

85 

125 

49 

94 

28 

37 

74 

156 

51 

97 

29 

33 

64 

186 

63 

99 

30 

8.  509  4130 

8. 511  7353 

1. 12217 

2.  3054 

5.  8702 

31 

26 

43 

248 

66 

05 

32 

23 

32 

278 

58 

08 

33 

19 

22 

'  309 

60  . 

10 

34 

16 

11 

340 

61 

13 

35 

12 

01 

370 

63 

16 

36 

08 

8.  511  7290 

401 

65 

19 

37 

05 

80 

482 

67 

22 

38 

01 

69 

462 

69 

24 

39 

8. 509  4098 

58 

493 

70 

27 

40 

8.509  4094 

8. 511  7248 

1.12523 

2. 3072 

5.  8730 

41 

91 

37 

554 

74 

33 

42 

87 

27 

584 

76 

35 

43 

84 

16 

615 

77 

38 

44 

80 

06 

646 

79 

41 

45 

77 

8. 511  7195 

676 

81 

44 

46 

73 

84 

707 

83 

46 

47 

70 

74 

737 

84 

49 

48 

66 

63 

768 

86 

52 

49 

63 

53 

798 

88 

55 

50 

8.  509  4059 

8. 511  7142 

1.12829 

2.  3090 

5.  8757 

51 

56 

31 

859 

91 

60 

.52 

52 

21 

889 

93 

63 

53 

49 

10 

920 

95 

66 

54 

45 

00 

950 

96 

69 

55 

41 

8.511  7089 

981 

98 

72 

56 

38 

78 

1. 13011 

2.  3100 

74 

57 

34 

68 

■  041 

02 

77 

58 

31 

57 

072 

03 

80 

59 

27 

46 

102 

05 

83 

60 

8. 509  4024 

8.  511  7036 

1. 13132 

2.3107 

5.  8785 

FACTORS  FOE  COMPUTATION  OF  GEODETIC  POSITIONS.        193 


Table  XXVII. — Factors  for  ilie  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  28°. 


log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

Latitude. 

diff.  1"=— 0.06 

difif.  l"=-0.1if 

diflf.  l"=+0.50 

diff.  l"=+0.03 

diff.  l"=+0.05 

28  00 

8. 509  4024 

8. 511  7036 

1. 13132 

2.  3107 

5. 8785 

1 

20 

25 

163 

09 

88 

2 

17 

14 

193 

10 

91 

3 

13 

04 

223 

12 

94 

4 

10 

8.  511  6993 

254 

14 

97 

05 

06 

82 

284 

15 

99 

6 

02 

72  . 

314 

17 

5.  8802 

7 

8.509  3999 

61 

345 

19 

05 

8 

95 

50 

375 

20 

08 

9 

92 

40 

405 

22 

11 

10 

8. 509  3988 

8. 511  6929 

1. 13435 

2.  3124 

5. 8813 

11 

85 

18 

465 

26 

16 

12 

81 

08 

496 

27 

19 

13 

78 

8. 511  6897 

526 

29 

22 

14 

74 

86 

556 

31 

25 

15 

70 

75 

586 

32 

27 

16 

67 

65 

616 

34 

30 

17 

63 

54 

646 

36 

33 

18 

60 

43 

677 

37 

36 

19 

56 

33 

707 

39 

39 

20 

8.509  3952 

8.511  6822 

1. 13737 

2.3141 

5.8841 

21 

49 

11 

767 

42 

44 

22 

45 

00 

797 

44 

47 

23 

42 

8.511  6790 

827  . 

46 

50 

24 

38 

79 

857 

47 

53 

25 

35 

68 

887 

49 

55 

26 

31 

57 

917 

51 

58 

27 

27 

47 

947 

52 

61 

28 

24 

36 

977 

54 

64 

29 

20 

25 

1. 14007 

56 

67 

30 

8.509  3917 

8.511  6714 

1. 14037 

2.  3157 

5.8870 

31 

13 

04 

067 

69 

72 

32 

09 

8. 511  6693 

097 

61 

75 

33 

06 

82 

127 

62 

78 

34 

02 

71 

157. 

64 

81 

35 

8.509  3899 

61 

187 

66 

84 

36 

95 

50 

217 

67 

87 

37 

92 

39 

247 

69 

89 

38 

83 

28 

277 

70 

92 

39 

84 

17 

307 

72 

95 

40 

8. 509  3881 

8. 511  6607 

1. 14337 

2. 3174 

5. 8898 

41 

77 

8.  511  6596 

366 

75 

5.8901 

42 

73 

85 

396 

77 

04   , 

43 

70 

74 

426 

79 

06 

44 

66 

63 

456 

80 

09 

45 

63 

52 

486 

82 

12 

46 

59 

42 

516 

83 

15 

47 

55 

31 

545 

85 

18 

48 

52 

20 

575 

87 

21 

49 

48 

09 

605 

88 

23 

50 

8. 509  3845 

8.  511  6498 

1. 14635 

2. 3190 

5.8926 

51 

41 

87 

664 

92 

29 

52 

37 

76 

694 

93 

32 

63 

34 

66 

724 

95 

35 

54 

30 

55 

754 

96 

38 

55 

26 

a 

783 

98 

40 

56 

23 

33 

813 

2. 3200 

43 

57 

19 

22 

843 

01 

46 

58 

16 

11 

872 

03 

49 

59 

12 

00 

902 

04 

52 

60 

8.509  3808 

8.511  6389 

1. 14932 

2.  3206 

5.8955 

MON   SXII- 


-13 


194 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  compatation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  29°. 


Latitude. 

log.  A 

log.  B 

log.C 

log.D 

log.E 

difif.  1"=— 0.  06 

difl.l"=— 0.18 

diff.l"=+0.49 

diff.  l"=+0. 03 

diff.l"=+0.05 

29  00 

8. 509  3808 

8.  511  6389 

1.14932 

2.3206 

5.8955 

1 

05 

78 

961 

08 

58 

01 

68 

991 

09 

60 

3 

8.509  3797 

57 

1.15021 

11 

63 

4 

94 

46 

050 

12 

66 

05 

90 

35 

080 

14 

69 

6 

86 

24 

109 

15 

72 

7 

83 

13 

139 

17 

75 

8 

79 

02 

168 

19 

78 

9 

76 

8. 511  6291 

198 

20 

80 

10 

8. 509  3772 

8.511  6280 

1. 15228 

2.3222 

5.8983 

11 

68 

69 

257 

23 

86 

12 

65 

58 

287 

25 

89 

13 

61 

47 

316 

26 

92 

14 

57 

36 

346 

28 

95 

15 

54 

26 

375 

30 

98 

16 

50 

15 

405 

31 

5.  9000 

17 

46 

04 

434 

33 

03 

18 

43 

8.  511  6193 

464 

34 

06 

19 

39 

82 

493 

36 

09 

20 

8.  509  3735 

8.  511  6171 

1.15522 

2.3237 

5.9012 

21 

32 

60 

552 

39 

15 

22 

28 

49 

581 

40 

18 

23 

24 

38 

611 

42 

21 

24 

21 

27 

640 

43 

23 

25 

17 

16 

670 

45 

26 

26 

13 

05 

699 

47 

29 

27 

10 

8.511  6094 

728 

48 

32 

28 

06 

83 

758 

50 

35 

29 

02 

72 

787 

51 

38 

30 

8. 509  3699 

8.  511  6061 

1.15816 

2.3253 

5.  9041 

31 

95 

50 

846 

54 

43 

32 

91 

39 

875 

56 

46 

33 

88 

28 

904 

57 

49 

34 

84 

17 

934 

59 

52 

35 

80 

06 

963 

60 

55 

36 

77 

8.  511  5995 

992 

62 

58 

37 

73 

84 

1. 16021 

63 

61 

38 

69 

73 

051 

65 

64 

39 

66 

61 

080 

66 

67 

40 

8. 509  3662 

8.  511  5950 

1. 16109 

2.  3268 

5. 9069 

41 

58 

39 

138 

69 

72 

42 

55 

28 

167 

71 

75 

43 

51 

17 

197 

72 

78 

44 

47 

06 

226 

74 

81 

45 

44 

8.  511  5895 

255 

75 

84 

46 

40 

84 

284 

77 

87 

47 

36 

73 

313 

78 

90 

48 

33 

62 

343 

80 

93 

49 

29 

51 

372 

81 

96 

50 

8.  509  3625 

8.  511  5840 

1. 16401 

2.  3283 

5. 9098 

51 

21 

29 

430 

84 

5.  9101 

52 

18 

18 

459 

86 

04 

53 

14 

06 

488 

87 

07 

54 

10 

8.  511  5795 

517 

89 

10 

55 

07 

84 

546 

90 

13 

56 

03 

73 

575 

92 

16 

57 

8.  509  3599 

62 

604 

93 

19 

58 

96 

51 

633 

95 

22 

59 

92 

40 

663 

96 

25 

60 

8.  509  3588 

8.511  5729 

1. 16692 

2.  3298 

5.  9127 

FACTORS  FOR  COMPUTATIOlSr  OF  GEODETIC  POSITIONS.        195 

Table  XXVIl. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  30°. 


log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

Latitude. 

dlff.l"  =  — 0.06 

diff.l"=— 0.19 

diff.l"=  +  0.48 

diff.l"  =  +0.02 

diff.  l"  =  +0.05 

30  00 

8.  509  3588 

8. 511  5729 

1. 16692 

2,  3298 

5. 9127 

1 

84 

18 

721 

99 

30 

2 

81 

06 

750 

2.  3301 

33 

3 

77 

8.511  5695 

778 

02 

36 

4 

73 

84 

807 

04 

39 

05 

69 

73 

836 

05 

42 

6 

66 

62 

865 

06 

45 

7 

62 

51 

894 

08 

48 

8 

58 

40 

923 

09 

51 

9 

55 

28 

952 

11 

54 

30 

8. 509  3551 

8.511  5617 

1. 16981 

2.  3312 

5.  9157 

11 

47 

06 

1. 17010 

14 

59 

12 

43 

8.511  5595 

039 

15 

62 

13 

40 

84 

068 

17 

65 

14 

36 

73 

097 

18 

68 

15 

32 

61 

126 

19 

71 

16 

29 

50 

155 

21 

74 

17 

25 

39 

184 

22 

77 

18 

21 

28 

212 

24 

80 

19 

17 

17 

241 

25 

83 

20 

8.  509  3514 

8.511  5505 

1. 17270 

2. 3327 

5.  9186 

21 

10 

8. 511  5494 

299 

28 

89 

22 

06 

83 

328 

30 

92 

23 

02 

72 

357 

31 

95 

24 

8.  509  3499 

61 

385 

32 

98 

25 

95 

49 

414 

34 

5. 9200 

26 

91 

38 

443 

35 

03 

27 

88 

27 

472 

37 

06 

28 

84 

16 

500 

38 

09 

29 

80 

04 

529 

39 

12 

30 

8.  509  3476 

8.511  5393 

1. 17558 

2.3341 

5.  9215 

31 

72 

82 

587 

42 

18 

32 

69 

71 

615 

44 

21 

33 

65 

59 

644 

45 

24 

34 

61 

48 

673 

47 

27 

35 

57 

37' 

701 

48 

30 

36 

54 

26 

730 

49 

33 

37 

50 

14 

759 

51 

36 

38 

46 

03 

788 

52 

39 

39 

42 

8.  511  5292 

816 

54 

42 

40 

8. 509  3439 

8. 511  5281 

1. 17845 

2.  3355 

5.9245 

41 

35 

69 

874 

56 

'48 

42 

31 

58 

902 

58 

51 

43 

27 

47 

931 

59 

53 

44 

24 

35 

959 

60 

56 

45 

20 

24 

988 

62 

59 

46 

16 

13 

1. 18017 

63 

62 

47 

12 

02 

045 

65 

65 

48 

09 

8.511  5190 

074 

66 

68 

49 

05 

79 

102 

67 

71 

50 

8.  509  3401 

8.511  5168 

1. 18131 

2.3368 

5.  9274 

51 

8. 509  3397 

56 

160 

70 

77 

52 

94 

45 

188 

71 

80 

53 

90 

34 

217 

73 

83 

54 

86 

22 

245 

74 

86 

55 

.82 

11 

274 

76 

89 

56 

78 

00 

302 

77 

92 

57 

75 

8.511  5088 

331 

78 

95 

58 

71 

77 

359 

80 

98 

59 

67 

66 

388 

81 

5.  9301 

60 

8. 509  3363 

8.511  5054 

1. 18416 

2.3382 

5. 9304 

196 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimufhs — Coutiuued. 

LATITUDE  31°. 


log.  A 

log.B 

log.  C 

log.  D 

log.  E 

Latitude. 

diff.  1"=— 0.06 

diff.  1"=— 0.19 

diff.l"=+0.47 

diff.l"=+0.02 

diff.l"=+0.05 

31  00 

8. 509  3363 

8. 511  5054 

1. 18416 

2. 3382 

5.9304 

1 

60 

43 

445 

84 

07 

2 

66 

32 

473 

85 

10 

3 

52 

20 

501 

86 

13 

4 

48 

09 

530 

88 

16 

05 

44 

8. 511  4998 

558 

89 

19 

6 

41 

86 

587 

90 

22 

7 

37 

75 

615 

92 

26 

8 

33 

61 

643 

93 

28 

9 

29 

52 

672 

95 

31 

10 

8. 509  3325 

8.  5U  4941 

1. 18700 

2. 3396 

5.9334 

11 

22 

29 

729 

97 

37 

12 

18 

18 

757 

99 

39 

13 

14 

07 

785 

2.  3400 

42 

14 

10 

8. 511  4895 

813 

01 

45 

15 

06 

84 

842 

02 

48 

16 

03 

72 

870 

04 

61 

17 

8.  509  3299 

61 

898 

06 

54 

18 

95 

50 

927 

06 

67 

19 

91 

38 

955 

08 

60 

20 

8.509  3287 

8.  511  4827 

1. 18983 

2.  3409 

5.  9363 

21 

84 

15 

1. 19012 

10 

66 

22 

80 

04 

040 

12 

69 

23 

76 

8. 611  4793 

068 

13 

72 

24 

81 

096 

14 

75 

25 

68 

70 

125 

16 

78 

26 

65 

58 

153  . 

17 

81 

27 

61 

47 

181 

18 

84 

28 

57 

35 

209 

20 

87 

29 

53 

24 

238 

21 

90 

30 

8. 509  3249 

8.  511  4713 

1. 19266 

2.  3422 

5.9393 

31 

46 

01 

294 

23 

96 

32 

42 

8.  511  4690 

322 

25 

99 

32 

38 

78 

351 

26 

5.  9402 

34 

34 

67 

379 

27 

05 

35 

30 

55 

407 

29 

08 

36 

26 

44 

435 

30 

11 

37 

23 

32 

463 

31 

14 

38 

19 

21 

491 

33 

17 

39 

15 

09 

520 

34 

20 

40 

8.509  3211 

8.  511  4598 

1. 19548 

2.3435 

6.  9423 

41 

07 

86 

576 

36 

26 

42 

03 

75 

604 

38 

29 

43 

00 

63 

632 

39 

32 

44 

8.  509  3196 

52 

660 

40 

35 

45 

92 

40 

688 

41 

38 

46 

83 

29 

716 

43 

41 

47 

84 

17 

744 

44 

44 

48 

81 

06 

772 

45 

47 

49 

77 

8. 511  4494 

800 

47 

50 

50 

8.  509  3173 

8.  511  4483 

1. 19828 

2.3448 

6.9453 

51 

09 

71 

856 

49 

56 

52 

65 

60 

884 

50 

69 

53 

61 

48 

912 

52 

62 

54 

67 

37 

940 

53 

65 

55 

54 

26 

968 

54 

68 

56 

50 

14 

996 

55 

72 

57 

46 

02 

1.  20024 

57 

75 

68 

42 

8.511  4391 

052 

58 

78 

59 

38 

79 

080 

59 

81 

60 

8. 509  3134 

8.5U  4368 

1.20108 

2.3460 

5.9484 

FACTORS  FOE  COMPUTATION"  OP  GEODETIC  POSITIONS. 


197 


Table  XXVll.— Factors  for 


;  compiitaUoii  of  geodetic  latitudes.  Ion 
LATITUDE  321 


\id  azimuths — Continued. 


log.  A 

log.B 

log.C 

log.D 

log.E 

Latitude. 

diff.  1"=— 0.06 

diff.  1"=— 0.19 

diff.  l"=+0.46 

dift'.  l"=+0.02 

diff.  l"=+0.05 

32  00 

8.509  3134 

8. 511  4368 

1.20108 

2.3460 

6. 9484 

1 

31 

56 

136 

62 

87 

27 

44 

164 

63 

90 

3 

23 

S3 

192 

64 

93 

4 

19 

21 

220 

66 

96 

05 

15 

10 

243 

67 

99 

6 

11 

8.511  4298 

276 

68 

6. 9502 

7 

07 

87 

304 

69 

05 

8 

04 

75 

332 

70 

08 

9 

00 

63 

360 

71 

11 

10 

8.  509  3096 

8.  511  4252 

1.  20337 

2.  3473 

5. 9514 

11 

92 

40 

415 

74 

17 

12 

38 

29 

443 

76 

20 

13 

84 

17 

471 

76 

23 

14 

80 

05 

499 

78 

26 

15 

76 

8.511  4194 

627 

79 

29 

16 

73 

82 

655 

80 

32 

17 

67 

71 

682 

.81 

36 

18 

65 

59 

610 

82 

38 

19 

61 

47 

638 

84 

41 

20 

8.509  3057 

8.  511  4136 

1. 20666 

2. 3485 

5. 9644 

21 

53 

24 

694 

36 

47 

22 

49 

13 

722 

87 

50 

23 

46 

01 

749 

88 

53 

24 

42 

8.511  4089 

777 

90 

66 

25 

38 

78 

805 

91 

60 

26 

34 

66 

833 

92 

63 

27 

30 

54 

860 

93 

06 

28 

26 

43 

888 

94 

69 

29 

22 

31 

916 

96 

72 

30 

8.  509  3018 

8. 611  4020 

1.  20944 

2. 3497 

5.  9575 

31 

15 

03 

971 

98 

78 

32 

11 

3. 511  3996 

999 

99 

81 

33 

07 

35 

1.  21027 

2. 3500 

84 

34 

03 

73 

054 

02 

87 

35 

8.509  2999 

61 

032 

03 

90 

36 

95 

50 

110 

04 

93 

37 

91 

33 

137 

05 

96 

33 

87 

26 

165 

06 

99 

39 

83 

15 

193 

07 

5.  9602 

40 

8.  509  2930 

3. 511  3903 

1.  21220 

2.  3509 

5.  9605 

41 

76 

8. 511  3391 

248 

10 

08 

42 

72 

79 

276 

11 

U 

43 

68 

63 

303 

12 

16 

44 

64 

56 

331 

13 

18 

45 

60 

44 

368 

14 

21 

46 

66 

33 

386 

16 

24 

47 

62 

21 

414 

17 

27 

43 

43 

09 

441 

18 

30 

49 

44 

8. 511  3798 

469 

19 

33 

50 

8. 509  2940 

8. 611  3786 

1. 21496 

2.  3620 

6.9636 

51 

37 

74 

524 

21 

39 

52 

33 

63 

551 

23 

42 

53 

29 

61 

579 

24 

46 

54 

25 

39 

607 

25 

48 

55 

21 

27 

634 

26 

51 

56 

17 

16 

662 

27 

64 

57 

13 

04 

689 

28 

68 

68 

09 

8. 511  3692 

717 

29 

61 

59 

06 

80 

744 

31 

64 

60 

8.  509  2901 

8. 611  3669 

1. 21772 

2. 3532 

6.  9667 

15:18 


A  MAXUAL  OF  TOPOGEAPHIO  METHODS. 


Table  XX^'II. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  38°, 


log.  A 

log.B 

log.  C 

log.  D 

log.  E 

Latitude. 

diff.l"=  — 0.07 

di£f.l"=  — 0.20 

diff.l"=+0.45 

AiS.  l"=+0.02 

diff.l"=+0.05 

33  00 

8.509  2901 

8.511  3669 

1.21772 

2.  3532 

5.  9667 

1 

8.509  2897 

57 

799 

33 

70 

2 

94 

45 

827 

34 

73 

3 

90 

33 

854 

35 

76 

i 

86 

22 

882 

36 

79 

05 

82 

10 

909 

37 

82 

6 

78 

8.  511  3598 

937 

38 

85 

7 

74 

86 

964 

40 

88 

8 

70 

75 

992 

41 

92 

9 

66 

63 

1.22019 

42 

95 

10 

8.509  2862 

8.  511  3551 

1.  22047 

2.  3643 

6. 9698 

11 

58 

39 

074 

44 

5.  9701 

12 

54 

28 

101 

45 

04 

13 

51 

16 

129 

46 

07 

14 

47 

04 

156 

47 

10 

15 

43 

8.611  3492 

184 

49 

13 

16 

39 

80 

211 

50 

16 

17 

35 

69 

238 

51 

19 

18 

31 

57 

266 

52 

22 

19 

27 

45 

293 

53 

26 

20 

8.  509  2823 

8.  511  3433 

1.  22321 

2.  3554 

6.  9729 

21 

19 

21 

348 

65 

32 

22 

15 

10 

375 

56 

35 

23 

11 

8.511  3398 

403 

57 

38  • 

24 

07 

86 

430 

58 

41 

25 

03 

74 

457 

60 

44 

26 

8.  509  2799 

62 

485 

61 

47 

27 

95 

51 

512 

62 

50 

28 

91 

39 

539 

63 

53 

29 

88 

27 

567 

64 

57 

30 

8.  509  2784 

8. 511  3315 

1. 22594 

2.  3565 

5.9760 

31 

80 

03 

621 

66 

63 

82 

76 

8.511  3291 

648 

67 

66 

33 

72 

80 

676 

68 

69 

34 

68 

68 

703 

69 

72 

35 

64 

58 

730 

70 

75 

36 

60 

44 

757 

71 

78 

37 

56 

32 

785 

73 

81 

38 

52 

20 

812 

74 

85 

39 

43 

09 

839 

75 

88 

40 

8.  509  2744 

8.  511  3197 

1.22866 

2.  3576 

5.  9791 

41 

40 

85 

893 

77 

94 

42 

36 

73 

921 

78 

97 

43 

32 

61 

948 

79 

5.9800 

44 

28 

49 

975 

80 

03 

45 

24 

37 

1. 23002 

81 

06 

40 

20 

25 

029 

82 

10 

47 

16 

13 

057 

83 

13 

48 

12 

02 

084 

84 

16 

49 

08 

8.511  3090 

111 

85 

19 

50 

8. 509  2704 

8.  511  3078 

1.  23138 

2.3586 

6.9822 

51 

01 

66 

165 

87 

26 

52 

8.509  2697 

54 

192 

88 

28 

53 

93 

42 

220 

89 

31 

54 

89 

30 

247 

91 

35 

55 

85 

18 

274 

92 

38 

56 

81 

06 

301 

93 

41 

57 

77 

8. 511  2995 

328 

94 

44 

68 

73 

83 

355 

95 

47 

59 

69 

71 

382 

96 

50 

60 

8. 509  2665 

8.  511  2959 

1.23409 

2.  3597 

6.9853 

FAOTOES  FOE  COMPUTATION  OF  GEODETIC  POSITIONS. 


199 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitades,  and  azimuths — Continued. 

LATITUDE  340. 


log.  A 

log.B 

lug.C 

log.B 

log.E 

Latitude 

diff.  1"=— 0.07 

diff.l"=_0.20 

diff,  1"  =  + 0.45 

difi'.  l"=+0.02 

diff.  1"  =  + 0.05 

34  00 

8. 509  2665 

8. 511  2959 

1. 23409 

2. 3697 

5.  9853 

1 

61 

47 

437 

98 

57 

2 

57 

35 

464 

99 

60 

3 

53 

23 

491 

2. 3600 

63 

4 

49 

11 

518 

01 

66 

05 

45 

8.511  2899 

546 

02 

69 

6 

41 

87 

572 

03 

72 

7 

37 

76 

699 

04 

76 

8 

33 

63 

626 

05 

79 

9 

30 

51 

653 

06 

82 

10 

8. 609  2625 

8.511  2840 

1. 23680 

2,  3607 

5. 9886 

11 

21 

28 

707 

08 

88 

12 

17 

16 

734 

09 

91 

13 

13 

04 

761 

10 

94 

14 

09 

8. 511  2792 

788 

11 

97 

15 

05 

80 

815 

12 

6. 9901 

16 

01 

68 

842 

13 

04 

17 

8. 609  2597 

56 

869 

14 

07 

18 

93 

44 

896 

16 

10 

19 

89 

32 

923 

16 

13 

20 

8.  609  2685 

8. 611  2720 

1. 23950 

2.  3617 

5. 9916 

21 

81 

08 

977 

18 

19 

22 

77 

8.  511  2696 

1.  24004 

19 

23 

23 

73 

84 

031 

20 

26 

24 

69 

72 

068 

21 

29 

25 

65 

60 

085 

22 

32 

26 

61 

48 

112 

23 

35 

27 

57 

36 

139 

24 

38 

28 

63 

24 

166 

25 

42 

29 

49 

12 

192 

26 

45 

30 

8. 509  2545 

8. 611  2600 

1.  24219 

2.  3627 

5.  9948 

31 

41 

8. 611  2688 

246 

28 

61 

32 

37 

76 

273 

29 

64 

33 

33 

64 

300 

30 

57 

34 

29 

62 

327 

31 

61 

35 

25 

40 

354 

32 

64 

36 

21 

28 

381 

33 

67 

37 

17 

16 

408 

34 

70 

38 

13 

04 

434 

36 

73 

39 

09 

8. 511  2492 

461 

36 

76 

40 

8. 509  2506 

8. 511  2480 

1.24488 

2.  3637 

5.  9980 

41 

0] 

68 

515 

38 

83 

42 

8.509  2497 

56 

542 

39 

86 

43 

93 

44 

569 

40 

89 

44 

89 

32 

595 

41 

92 

45 

85 

20 

622 

42 

96 

46 

81 

08 

649 

43 

99 

47 

77 

8.511  2396 

676 

44 

6. 0002 

48 

73 

84 

703 

44 

06 

49 

69 

72 

729 

45 

08 

50 

8.509  2465 

8.511  2360 

1.  24756 

2.  3646 

6. 0011 

51 

61 

48 

783 

47 

15 

52 

67 

35 

810 

48 

18 

53 

53 

23 

837 

49 

21 

54 

49 

11 

863 

50 

24 

55 

45 

8. 511  2299 

890 

51 

27 

5S 

41 

87 

917 

52 

31 

67 

37 

76 

944 

63 

34 

58 

33 

63 

970 

64 

37 

59 

29 

51 

997 

66 

40 

60 

8. 509  2425 

2.511  2239 

1.  26024 

2. 3666 

6. 0043 

200 


A  MANUAL  OP  TOPOGEAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  comjyutation  of  geodetic  latitudes,  longitudes,  and  azimuths — Coutinued. 

LATITtTDE  S5°. 


log.  A 

log.B 

log.C 

log.D 

log.E 

Latitude. 

diff.  1"=— 0.07 

diff.  1"=— 0.20 

diff.  1"=  +  0.44 

difl',l"=  +  0.01 

diff.  1"=  +  0.05 

35  00 

8.509  2425 

8.  511  2239 

1.  25024 

2.  3656 

6,  0043 

1 

21 

27 

050 

57 

47 

2 

17 

15 

077 

68 

50 

3 

13 

03 

104 

59 

53 

4 

09 

8.5112191 

131 

59 

56 

05 

05 

78 

157 

60 

59 

6 

01 

66 

184 

61 

63 

7 

8..=;09  2396 

54 

211 

62 

66 

8 

93 

42 

237 

63 

69 

9 

88 

30 

264 

64 

72 

10 

8.509  2384 

8. 511  2118 

1.25291 

2.3665 

6.  0075 

11 

80 

06 

317 

66 

79 

12 

76 

8.  511  2094 

344 

67 

82 

13 

72 

82 

371 

68 

85 

14 

68 

70 

397 

69 

88 

15 

64 

57 

424 

70 

91 

16 

60 

45 

461 

70 

96 

17 

56 

33 

477 

71 

98 

18 

52 

21 

504 

72 

6.  0101 

19 

48 

09 

531 

73 

04 

20 

8.509  2344 

8.511  1997 

1.  25557 

2.3674 

6.  0107 

21 

40 

85 

584 

76 

11 

22 

36 

72 

610 

76 

14 

23 

32 

60 

637 

77 

17 

24 

28 

48 

664 

78 

20 

25 

24 

36 

690 

79 

23 

26 

20 

24 

717 

79 

27 

27 

16 

12 

743 

80 

30 

28 

12 

00 

770 

81 

33 

29 

08 

8.  611  1887 

796 

82 

36 

30 

8.509  2304 

8.  511  1875 

1. 25823 

2.3683 

6. 0140 

31 

00 

63 

850 

84 

43 

32 

8. 609  2296 

51 

876 

85 

46 

33 

92 

39 

903 

86 

49 

34 

87 

27 

929 

86 

52 

35 

83 

15 

956 

87 

56 

36 

79 

02 

982 

88 

59 

37 

75 

8.511  1790 

1.26009 

89 

62 

38 

71 

78 

036 

90 

65 

39 

67 

66 

062 

91 

69 

40 

8.609  2263 

8.  511  1754 

1. 26088 

2. 3692 

6.  0172 

41 

59 

41 

115 

93 

75 

42 

65 

29 

141 

93 

78 

43 

51 

17 

168 

94 

81 

44 

47 

05 

194 

95 

85 

45 

43 

8.  511  1693 

221 

96 

88 

46 

39 

80 

247 

97 

91 

47 

35 

68 

274 

98 

94 

48 

31 

56 

300 

99 

98 

49 

27 

44 

327 

99 

6.  0201 

50 

8.  509  2222 

8.  511  1632 

1. 26353 

2.  3700 

6.0204 

51 

18 

20 

380 

01 

07 

52 

14 

07 

406 

02 

11 

53 

10 

8.511  1596 

432 

03 

14 

54 

06 

83 

469 

04 

17 

55 

02 

71 

485 

05 

20 

56 

8. 509  2198 

58 

512 

05 

24 

57 

94 

46 

538 

06 

27 

58 

90 

34 

665 

07 

30 

59 

86 

22 

591 

08 

33 

60 

8.  509  2182 

8.  511  1510 

1.26617 

3.  3709 

6.  0237 

FACTOES  FOE  COMPUTATION  OF  GEODETIC  POSITIONS.       201 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  Jongitudes,  and  azimuths — Continued. 

LATITUDE  36° 


log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

Latitude. 

dlflf.l"  =  -0.07 

diff.  1"=— 0.20 

diff.  l"=+0.44 

diff.  J"=+0.01 

diff.  ]  "=+0.05 

36  00 

8.  509  2182 

8.  511  1510 

1. 26617 

.   2.3709 

6.0237 

1- 

78 

8.511  1497 

644 

10 

40 

2 

74 

85 

670 

10 

43 

3 

70 

73 

697 

11 

46 

4 

65 

61 

723 

12 

60 

05 

61 

48 

749 

13 

53 

6 

57 

36 

776 

14 

56 

7 

53 

24 

802 

14 

59 

8 

49 

12 

828 

15 

63 

9 

45 

8.511  1399 

855 

16 

66 

10 

8. 509  2141 

8. 511  1387 

1.  26881 

2.  3717 

6.0269 

11 

37 

75 

908 

18 

72 

12 

33 

63 

934 

19 

76 

13 

29 

50 

960 

19 

79 

14 

25 

38 

987 

20 

82 

15 

21 

26 

1.  27013 

21 

85 

16 

16 

14 

039 

22 

89 

17 

12 

01 

066 

23 

92 

18 

08 

8.511  1289 

092 

23 

95 

19 

04 

77 

118 

24 

99 

20 

8.  509  2100 

8. 511  1265 

1. 27145 

2.  3725 

6.  0302 

21 

8. 509  2096 

52 

171 

26 

05 

22 

92 

40 

197 

27 

08 

23 

88 

28 

223 

27 

12 

24 

84 

15 

250 

28 

15 

25 

80 

03 

276 

29 

18 

26 

75 

8.511  1191 

302 

30 

21 

27 

71 

79 

329 

31 

25 

28 

67 

66 

355 

31 

28 

29 

63 

54 

381 

32 

31 

30 

8.  509  2059 

8.511  1142 

1.  27407 

2.  3733 

6. 0334 

31 

55 

29 

434 

34 

38 

32 

51 

17 

460 

35 

41 

33 

47 

05 

486 

35 

44 

34 

43 

8.  511  1092 

512 

36 

48 

35 

39 

80 

639 

37 

51 

36 

35 

68 

565 

38 

64 

37 

30 

56 

591 

38 

57 

38 

26 

43 

617 

39 

61 

39 

22 

31 

644 

40 

64 

40 

8. 509  2018 

8.  511  1019 

1. 27670 

2.  3741 

6.  0367 

41 

14 

06 

696 

41 

71 

42 

10 

8. 511  0994 

722 

42 

74 

43 

06 

82 

748 

43 

77 

44 

02 

69 

775 

44 

80 

46 

8.509  1998 

57 

801 

45 

84 

46 

93 

45 

827 

45 

87 

47 

89 

32 

853 

46 

90 

48 

85 

20 

879 

47 

94 

49 

81 

08 

905 

48 

97 

50 

8. 509  1977 

8. 511  0895 

1.  27932 

2.  3748 

6. 0400 

51 

73 

83 

958 

49 

03 

52 

69 

71 

984 

50 

07 

53 

65 

58 

1.  28010 

51 

10 

54 

61 

46 

036 

51 

13 

55 

56 

34 

062 

52 

17 

56 

52 

21 

088 

63 

20 

57 

48 

09 

114 

54 

23 

58 

44 

8.  611  0797 

141 

54 

27 

59 

40 

84 

167 

55 

30 

60 

8. 509  1936 

8.  511  0772 

1. 28193 

2. 3756 

6.  0433 

202 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  a^iniulhn — Continued. 

LATITUDE  87°. 


log.  A 

log.  B 

log.  C 

log.  D 

log.E 

Latitude. 

dUi".l"=— 0.07 

diff.l"=  — 0.21 

diir.l"=  +  0.43 

diff.  1"—  +  0. 01 

diff.  1"=  + 0.06 

37  00 

8.509  1936 

8. 511  0772 

1.28193 

2. 3756 

6.  0433 

1 

32 

60 

219 

56 

.   37 

28 

47 

245 

57 

40 

3 

23 

35 

271 

58 

43 

i 

19 

22 

297 

69 

46 

05 

15 

10 

324 

69 

50 

6 

11 

8.  .511  0698 

350 

60 

53 

07 

85 

376 

61 

56 

8 

03 

73 

402 

62 

60 

9 

8. 509  1899 

61 

428 

62 

63 

10 

8. 509  1895 

8.611  0648 

1.28454 

2. 3763 

6,  0466 

11 

90 

36 

480 

64 

70 

12 

86 

23 

506 

65 

73 

13 

82 

11 

532 

65 

76 

14 

78 

8.511  0599 

558 

66 

80 

15 

74 

86 

584 

67 

83 

16 

70 

74 

610 

67 

86 

17 

66 

61 

636 

68 

89 

18 

62 

49 

662 

69 

93 

19 

57 

37 

638 

69 

96 

20 

8.509  1853 

8.511  0524 

1.  28715 

2.  3770 

6. 0499 

21 

49 

12 

741 

71 

6.  0503 

22 

45 

00 

767 

72 

06 

23 

a 

8.511  0487 

793 

72 

09 

24 

37 

75 

819 

73 

13 

25 

33 

62 

.   845 

74 

16 

26 

28 

50 

871 

74 

19 

27 

24 

37 

897 

75 

23 

28 

20 

25 

923 

76 

26 

29 

16 

13 

949 

76 

29 

30 

8.509  1812 

8. 511  0400 

1. 28975 

2.377/ 

6. 0533 

31 

08 

8.  511  0388 

1.  29001 

78 

36 

32 

04 

75 

027 

79 

39 

33 

00 

63 

053 

79 

43 

34 

8.  509  1795 

51 

079 

80 

46 

35 

91 

38 

104 

81 

49 

36 

87 

26 

130 

81 

53 

37 

83 

13 

166 

82 

56 

38 

79 

01 

182 

83 

59 

39 

75 

8.511  0288 

208 

83 

63 

40 

8. 509  1771 

8.511  0276 

1.  29234 

2.  3784 

6.  0566 

41 

66 

64 

260 

85 

69 

42 

62 

51 

286 

86 

73 

43 

58 

39 

312 

86 

76 

44 

54 

26 

338 

87 

79 

45 

50 

14 

364 

87 

83 

46 

46 

01 

390 

88 

86 

47 

41 

8.  511  0189 

416 

89 

89 

48 

37 

76 

442 

89 

93 

49 

33 

64 

468 

90 

96 

50 

8. 509  1729 

8.  511  0151 

1.29494 

2,  3791 

6.  0600 

51 

25 

39 

620 

91 

03 

52 

21 

26 

546 

•i   92 

06 

53 

16 

14 

571 

93 

10 

54 

12 

02 

697 

93 

13 

55 

08 

8.  511  0089 

623 

94 

16 

56 

04 

77 

649 

95 

20 

57 

00 

64 

675 

95 

23 

58 

8.509  1696 

52 

701 

96 

26 

59 

92 

39 

727 

96 

30 

60 

8.  509  1687 

8.  511  0027 

1.  29753 

2.  3797 

6. 0633 

FAOTOES  FOli  COMPUTATION  OF  GEODETIC  POSITIONS.       203 

Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  38°. 


log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

Latitude. 

difl.l"=— 0.07 

aiff.l"=— 0.21 

diff.l"=  +  0.43. 

diff.l"  =  +  0.01 

diff.l"=  +  0.06 

38  00 

8.509  1687 

8.  511  0027 

1.  29753 

2.  3797 

6.  0633 

1 

83 

14 

778 

98 

36 

2 

79 

02 

804 

98 

40 

3 

75 

8. 510  9989 

830 

99 

43 

4 

71 

77 

856 

2.  3800 

47 

05 

67 

64 

882 

00 

50 

6 

62 

52 

908 

01 

53 

7 

58 

39 

934 

02 

57 

8  ■ 

54 

27 

959 

02 

60 

9 

50 

14 

985 

03 

63 

.0 

8.  509  1646 

8.  510  9902 

1.  30011 

2.  3803 

6. 0667 

11 

42 

8.  510  9889 

037 

04 

70 

12 

37 

77 

063 

05 

73 

13 

33 

64 

089 

05 

77 

14 

29 

52 

114 

06 

80 

16 

25 

39 

140 

07 

84 

16 

21 

27 

166 

07 

87 

17 

17 

14 

192 

08 

90 

18 

12 

02 

218 

08 

94 

19 

08 

8.510  9789 

243 

09 

97 

20 

8.509  1604 

8. 510  9777 

1.  30269 

2.  3810 

6. 0701 

21 

00 

64 

295 

10 

04 

22 

8.  509  1596 

52 

321 

U 

07 

23 

92 

39 

347 

12 

11 

24 

87 

-   27 

372 

12 

14 

25 

83 

14 

398 

13 

17 

26 

79 

01 

424 

13 

21 

27 

75 

8.  510  9689 

450 

14 

24 

28 

71 

77 

476 

16 

28 

29 

66 

64 

501 

15 

31 

30 

8.509  1562 

8.  510  9652 

1.30527 

2.  3816 

6.  0734 

31 

58 

39 

553 

16 

38 

32 

54 

27 

579 

17 

41 

33 

50 

14 

604 

17 

44 

34 

46 

01 

630 

18 

48 

35 

41 

8.510  9589 

656 

19 

51 

36 

37 

76 

682 

19 

66 

37 

33 

64 

707 

20 

58 

38 

29 

51 

733 

20 

61 

39 

25 

39 

769 

21 

65 

40 

8.  509  1521 

8.  510  9526 

1. 30785 

2.  3822 

6.  0768 

41 

16 

14 

810 

22 

72 

42 

12 

01 

836 

23 

75 

43 

08 

8.  510  9488 

862 

23 

78 

44 

04 

76 

88T 

24 

82 

45 

00 

63 

913 

24 

85 

46 

8. 509  1495 

61 

939 

25 

89 

47 

91 

38 

965 

26 

92 

48 

87 

26 

990 

26 

95 

49 

83 

13 

1.  31016 

27 

99 

50 

8. 509  1479 

8.  510  9401 

1. 31042 

2.  3827 

6.  0802 

51 

75 

8.  510  9388 

067 

28 

06 

52 

70 

76 

093 

28 

09 

53 

66 

63 

119 

29 

13 

54 

62 

50 

144 

30 

16 

55 

58 

38 

170 

30 

19 

56 

53 

25 

196 

31 

23 

57 

49 

13 

221 

31 

26 

58 

45 

00 

247 

32 

30 

59 

41 

8.510  9287 

273 

32 

33 

60 

8.  509  1437 

8.  510  9275 

1. 31299 

2.3833 

6. 0836 

•204 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  compntaiion  of  geodetic  latitudes,  longitudes,  and  azimuths — Contiuvied. 

LATITUDE  S90. 


log  A 

logB 

log  C 

log  I) 

logE 

Latitude. 

difif.  1"=— 0.07 

diff.  1"=— 0.21 

diff.  I"=+0.43 

diff.  1"=+0.01 

ditt.  l"=+0.06 

39  00 

8. 509  1437 

8.  510  9275 

1.  31299 

2. 3833 

6.  0836 

1 

33 

62 

324 

33 

40 

2 

28 

50 

350 

34 

43 

3 

24 

37 

375 

35 

47 

4 

20 

25 

401 

35 

50 

05 

16 

12 

427 

36 

53 

6 

12 

8.  510  9199 

452 

36 

57 

7 

07 

87 

478 

37 

60 

8 

03 

74 

504 

37 

-   64 

9 

8.509  1399 

62 

529 

38 

67 

10 

8.  509  1395 

8.  510  9149 

1.  31555 

2.3838 

6.0871 

U 

91 

36 

581 

39 

74 

12 

86 

24 

606 

39 

77 

13 

82 

11 

632 

2.3840 

81 

U 

78 

8. 510  9098 

658 

40 

84 

15 

74 

86 

683 

41 

88 

16 

70 

73 

709 

41 

91 

17 

65 

61 

734 

42 

95 

18 

61 

48 

760 

43 

98 

19 

57 

36 

786. 

43 

6.  0902 

20 

8. 509  1353 

8.  510  9023 

1.31811  ■ 

2.  3844 

6.  0905 

21 

49 

10 

837 

44 

08 

22 

44 

8.  510  8998 

862 

45 

12 

23 

40 

85 

888 

45 

15 

2i 

36 

73 

913 

46 

19 

25 

32 

60 

939 

46 

22 

26 

28 

47 

965 

47 

26 

27 

23 

35 

990 

47 

29 

28 

19 

23 

1.32016 

48 

32 

29 

15 

09 

041 

48 

38 

30 

8. 509  1311 

8.  510  8897 

1. 32067 

2.  3849 

6.0939 

31 

07 

84 

092 

49 

43 

32 

02 

72 

118 

2.3850 

46 

33 

8.  509  1298 

59 

144 

50 

50 

34 

S4 

46 

169 

51 

53 

35 

90 

34 

195 

51 

57 

36 

86 

21 

220 

52 

60 

37 

81 

08 

246 

52 

63 

38 

77 

8.510  8796 

271 

53 

67 

39 

73 

83 

297 

53 

70 

40 

8.509  1269 

8.510  8771 

1.  32323 

2. 3854 

6.0974 

41 

64 

58 

348 

54 

77 

42 

60 

45 

374 

55 

81 

43 

56 

33 

399 

55 

84 

44 

52 

20 

425 

56 

83 

45 

48 

07 

450 

56 

91 

46 

43 

8. 510  8695 

476 

57 

95 

47 

39 

82 

501 

57 

98 

48 

35 

69 

527 

57 

6. 1002 

49 

31 

57 

552 

58 

05 

50 

8.  509  1227 

8.510  8644 

1. 32578 

2.3858 

6. 1008 

51 

22 

31 

603 

59 

12 

52 

18 

19 

629 

59 

15 

53 

14 

06 

654 

2.3860 

19 

54 

10 

8. 510  8593 

680 

60 

22 

55 

06 

81 

705 

61 

26 

56 

01 

68 

731 

61 

29 

57 

8.509  1197 

55 

756 

62 

33 

58 

93 

43 

782 

62 

36 

59 

89 

30 

807 

63 

40 

60 

8.509  1184 

8. 510  8517 

1.  32833 

2. 3863 

6. 1043 

FACTOES  FOE  COMPUTATION  OF  GEODETIC  POSITIOE^S.        205 

Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  aaimutlis — Continued. 

LATITUDE  40°. 


log  A 

log  B 

log  C 

log  D 

log  E 

Latitude. 

diff.l"=— 0.07 

diff.l"=— 0.2'l 

diff.l"=+0.42 

diff.l"=  +  0.01 

diff.  1"=  +  0.06 

40  00 

8.  509  1184 

8.510  8517 

1. 32833 

2.  3863 

6.1043 

1 

80 

05 

858 

64 

47 

2 

76 

8. 510  8492 

884 

64 

50 

3 

72 

79 

909 

64 

54 

4 

67 

67 

935 

65 

57 

05 

63 

54 

960 

65 

61 

6 

59 

41 

986 

66 

64 

7 

55 

29 

1.  33011 

66 

67 

8 

60 

16 

037 

67 

71 

9 

46 

03 

062 

67 

74 

10 

8. 509  1142 

8.  510  8391 

1.  33688 

2.  3868 

6. 1078 

11 

38 

78 

113 

68 

81 

12 

34 

65 

139 

68 

85 

13 

29 

53 

164 

69 

88 

14 

25 

40 

189 

69 

92 

15 

21 

27 

215 

2.  3870 

95 

16 

17 

15 

240 

70 

99 

17 

12 

02 

266 

71 

6. 1102 

18 

OS 

8.  510  8289 

291 

71 

06 

19 

04 

77 

317 

72 

09 

20 

8.509  1100 

8. 510  8264 

1.  33342 

2.  3872 

6.1113 

21 

8.  509  1096 

51 

363 

72 

16 

22 

91 

38 

393 

73 

20 

23 

87 

26 

418 

73 

23 

24 

83 

13 

444 

74 

27 

25 

79 

00 

469 

74 

30 

26 

74 

8.510  8188 

495 

74 

34 

27 

70 

75 

520 

75 

37 

28 

66 

62 

546 

75 

41 

29 

62 

50 

571 

76 

44 

30 

8.509  1057 

8.  510  8137 

1.33596 

2. 3876 

6. 1148 

31 

53 

24 

622 

77 

51 

32 

49 

11 

■  647 

77 

55 

33 

45 

8. 510  8099 

673 

77 

58 

34 

41 

86 

698 

78 

62 

35 

36 

73 

723 

78 

65 

36 

32 

61 

749 

79 

69 

37 

28 

48 

774 

79 

72 

38 

24 

35 

800 

79 

76 

39 

19 

23 

825 

2.  3880 

79 

40 

8. 509  1015 

8.  510  8010 

1.  33850 

2.3880 

6. 1183 

41 

11 

8.510  7997 

876 

81 

86 

42 

07 

84 

901 

81 

90 

43 

02 

72 

926 

81 

93 

44 

8.509  0998 

59 

952 

82 

97 

45 

94 

46 

977 

82 

6. 1200 

46 

90 

33 

1.  34003 

83 

04 

47 

85 

21 

028 

83 

07 

48 

81 

08 

053 

83 

11 

49 

77 

8.  510  7895 

079 

84 

15 

50 

8.  509  0973 

8.  510  7883 

1.  34104 

2.3884 

6. 1218 

51 

68 

70 

129 

84 

22 

52 

64 

57 

155 

85 

25 

53 

60 

44 

180 

85 

29 

54 

56 

32 

206 

86 

32 

55 

52 

19 

231 

86 

36 

56 

47 

06 

256 

86 

39 

57 

43 

8. 510  7793 

282 

87 

43 

58 

39 

81 

307 

87 

46 

59 

34 

68 

332 

87 

50 

60 

8.509  0930 

8. 510  7755 

1. 34358 

2.  3888 

6. 1253 

206 


A  MANUAL  OP  TOPOGEAPHIO  METHODS. 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  41°. 


log.  A 

log.B 

log.C 

log.D 

log.E 

latitude. 

difF.l"=— 0.07 

diff.l"  =  — 0.21 

diff.  1"  =  +  0.42 

diff.  1"=  + 0.01 

diff.  1"=  + 0.06 

41  00 

8.  509  0930 

8.510  7755 

1. 34358 

2. 3888 

6. 1253 

1 

26 

42 

383 

88 

57 

2 

22 

30 

408 

89 

60 

3 

18 

17 

434 

89 

64 

4 

13 

04 

459 

89 

67 

05 

09 

8. 510  7691 

484 

90 

71 

6 

05 

79 

510 

90 

75 

7 

00 

66 

535 

90 

78 

8 

8.509  0896 

53 

560 

91 

82 

9 

92 

40 

586 

91 

85 

10 

8.509  0888 

8.  510  7628 

1.34611 

2.  3891 

6. 1289 

11 

83 

15 

636 

92 

92 

12 

79 

02 

662 

92 

96 

13 

75 

8. 510  7590 

687 

93 

99 

14 

71 

77 

712 

93 

6. 1303 

15 

67 

64 

738 

93 

06 

16 

62 

51 

763 

94 

10 

17 

58 

39 

788 

94 

14 

18 

54 

26 

814 

94 

17 

19 

49 

13 

839 

95 

21 

211 

8. 509  0845 

8.  510  7500 

1.34864 

2.3895 

6. 1324 

21 

41 

8.  510  7488 

890 

95 

28 

23 

37 

75 

915 

96 

31 

23 

32 

62 

940 

96 

35 

24 

28 

49 

965 

96 

38 

25 

24 

36 

991 

97 

42 

26 

20 

24 

1.  35016 

97 

46 

27 

15 

11 

041 

97 

49 

28 

11 

8.510  7398 

066 

98 

53 

29 

07 

85 

092 

98 

56 

30 

8. 509  0803 

8.  510  7373 

1. 35117 

2.  3898 

6. 1360 

31 

8.509  0798 

60 

142 

99 

63  - 

32 

94 

■  47 

168 

99 

67 

33 

90 

34 

193 

99 

70 

34 

86 

22 

218 

2.3900 

74 

35 

81 

09 

243 

00 

78 

36 

77 

8. 510  7296 

269 

00 

81 

37 

73 

83  ■ 

294 

00 

85 

38 

69 

70 

319 

01 

88 

39 

64 

58 

345 

01 

92 

40 

».  509  0760 

8.510  7245 

1.35370 

2.  3901 

6. 1395 

41 

56 

32 

395 

02 

99 

42 

52 

19 

420 

02 

6. 1403 

43 

47 

07 

446 

02 

06 

44 

43 

8.  510  7194 

471 

03 

10 

45 

39 

81 

496 

03 

13 

46 

35 

68 

522 

03 

17 

47 

30 

55 

547 

03 

20 

48 

26 

43 

572 

04 

24 

49 

22 

30 

597 

04 

28 

50 

8. 509  0738 

8.  510  7117 

1.  35623 

2.3904 

6. 1431 

51 

13 

04 

648 

05 

35 

52 

09 

8.  510  7091 

673 

OS 

38 

53 

05 

79 

698 

05 

42 

51 

00 

66 

723 

05 

46 

55 

8. 509  0696 

53 

749 

06 

49 

56 

92 

40 

774 

06 

53 

57 

88 

27 

799 

06 

56 

58 

83 

15 

824 

07 

60 

59 

79 

02 

850 

07 

63 

60 

8. 509  0675 

8.  510  6989 

1.35875 

2.  3907 

6. 1467 

FACTOES  FOE  COMPUTATION  OF  GEODETIC  POSITIONS.        2()7 


Table  XXVIT. — Factors  for  the  compntaUon  of  ffcodelic  latitudes,  longitudes,  and  azimuths — Contimied. 

LATITUDE  42° 


log.  A 

log.  B 

Ing.C 

logD. 

log.  E 

Latitude. 

diff.l"=— 0.07 

:difl'.  1"=— 0.21 

ditf.  l"=+0.42 

diff.  l"=+0.00 

diff.  1"  =+0.06 

42  00 

8. 509  0675 

8.  510  6989 

1.  35875 

2.3907 

6. 1467  • 

1 

71 

76 

900 

08 

71 

2 

66 

64 

925 

08 

74 

3 

62 

51 

951 

08 

78 

4 

58 

38 

976 

08 

81 

05 

54 

25 

1.  36001 

09 

85 

6 

49 

12 

026 

09 

89 

7 

45 

00 

052 

09 

92 

8 

41 

8. 510  6887 

077 

09 

96 

9 

36 

74 

102 

10 

99 

10 

8. 509  0632 

8.510  6861 

1.  36127 

2.  3910 

6. 1503 

11 

28 

48 

152 

10 

07 

12 

24 

36 

178 

10 

10 

13 

19 

23 

203 

11 

14 

14 

15 

10 

228 

11 

17 

15 

11 

8. 510  6797 

253 

11 

21 

16 

07 

84 

278 

12 

25 

17 

02 

72 

304 

12 

28 

18 

8. 509  0598 

59 

329 

12 

32 

19 

94 

46 

354 

12 

35 

20 

8.509  6590 

8.510  6733 

1.  36379 

2. 3913 

6. 1539 

21 

85 

20 

404 

13 

43 

22 

81 

07 

430 

13 

46 

23 

77 

8. 510  6695 

455 

13 

50 

24 

72 

82 

480 

13 

54 

25 

68 

69 

505 

14 

57 

26 

64 

56 

530 

14 

61 

27 

60 

43 

556 

14 

64 

28- 

55 

31 

681 

14 

68 

29 

51 

18 

606 

15 

72 

30 

8.509  0547 

8.530  6605 

1.36631 

2. 3915 

6. 1575 

31 

43 

8. 510  6592 

056 

15 

79 

32 

38 

79 

682 

15 

83 

33 

34 

66 

707 

16 

86 

84 

30 

54 

732 

16 

90 

35 

25 

41 

757 

16 

93 

36 

21 

28 

782 

16 

97 

37 

17 

15 

808 

17 

6. 1601 

38 

13 

02 

833 

17 

04 

39 

08 

8. 510  6490 

858 

17 

08 

40 

8. 509  0504 

8,  510  6477 

1. 36883 

2.  3917 

6. 1612 

41 

00 

64 

908 

17 

15 

42 

8. 509  0496 

51 

934 

18 

19 

43 

91 

38 

959 

18 

22 

44 

87 

25 

984 

18 

26 

45 

83 

13 

1. 37009 

18 

30 

46 

78 

00 

034 

19 

33 

47 

74 

8.510  6387 

0.59 

19 

37 

48 

70 

74 

085 

19 

41 

49 

66 

61 

110 

19 

44 

50 

8.  609  0461 

8.510  6348 

1. 37135 

2.  3919 

6. 1648 

51 

57 

36 

160 

20 

52 

52 

53 

23 

185 

20 

55 

53 

48 

10 

210 

20 

59 

54 

44 

8. 610  6297 

235 

20 

63 

55 

40 

84 

261 

20 

66 

56 

36 

71 

286 

21 

70 

57 

31 

59 

311 

21 

73 

68 

27 

46 

336 

21 

77 

59 

23 

33 

361 

21 

81 

60 

8.  509  0419 

8. 510  6220 

1. 37386 

2. 3921 

6. 1084 

208 


A  MANUAL  OP  TOPOGEAPHIO  METHODS. 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitiides,  longitudes,  and  azimuths — Contiuued. 

LATITDDE  43°. 


log.  A 

log.B 

log.C 

log.D 

log.E 

Latitude. 

diff.  1"=— 0.07 

diff.  1"«=— 0.21 

diff'.  1"=  + 0.42 

diff.  1"=  + 0.00 

diff.  1"=+ 0.06 

43  00 

8. 509  0419 

8.  510  6220 

1.37386 

2.  3921 

6. 1684 

1 

14 

07 

412 

22 

88 

2 

10 

8.  510  6195 

437 

22 

92 

3 

06 

82 

462 

22 

95 

i 

01 

69 

487 

22 

99 

05 

8. 509  0397 

56 

512 

22 

6, 1703 

6 

93 

43 

537 

22 

06 

7 

89 

30 

663  ■ 

23 

10 

8 

84 

17 

588 

23 

14 

9 

80 

05 

613 

23 

17 

10 

8. 509  0376 

8. 510  G092 

1.  37638 

2.3923 

6. 1721 

11 

71 

79 

663 

23 

25 

12 

67 

66 

688 

24 

28 

13 

63 

53 

713 

24 

32 

14 

59 

40 

739 

24 

36 

15 

54 

28 

764 

24 

39 

16 

50 

15 

789 

24 

43 

17 

46 

02 

814 

24 

47 

18 

41 

8.  510  5989 

839 

25 

50 

19 

37 

76 

864 

25 

54 

20 

8.  509  0333 

8. 510  5963 

1.  37889 

2.  3925 

6. 1758 

21 

29 

50 

915 

25 

61 

22 

24 

38 

940 

25 

65 

23 

20 

25 

965 

25 

69 

24 

16 

12 

990 

25 

72 

25 

12 

8.510  5899 

1.  38015 

26 

76 

26 

07 

86 

040 

26 

80 

27 

03 

73 

065 

26 

83 

28 

8.  509  0299 

60 

091 

26 

•  87 

29 

94 

48 

116 

26 

91 

30 

8. 509  0290 

8. 510  5835 

1.  38141 

2. 3926 

6. 1795 

31 

86 

22 

166 

27 

98 

32 

82 

09 

191 

27 

6. 1802 

33 

77 

8.510  6796 

216 

27 

06 

34 

73 

83 

241 

27 

09 

35 

69 

71 

266 

27 

13 

36 

64 

58 

292 

27 

17 

37 

60 

45 

317 

27 

20 

38 

56 

32 

342 

27 

24 

39 

52 

19 

367 

28 

23 

40 

8.  509  0247 

8. 510  .5706 

1.  38392 

2.3928 

6. 183] 

41 

43 

8.510  5693 

417 

28 

35 

42 

39 

81 

442 

28 

39 

43 

34 

68 

467 

28 

42 

44 

30 

55 

492 

28 

46 

45 

26 

42 

518 

28 

50 

46 

22 

29 

543 

28 

53 

47 

17 

16 

568 

29 

57 

48 

13 

03 

593 

29 

61 

49 

09 

8.510  5591 

618 

29 

65 

50 

8.609  0204 

8.  510  5578 

1.  38643 

2.  3929 

6. 1868 

51 

00 

65 

668 

29 

72 

52 

8.  509  0196 

52 

693 

29 

76 

53 

92 

39 

719 

29 

79 

54 

87 

26 

744 

29 

83 

55 

83 

13 

769 

30 

87 

56 

79 

01 

794 

30 

91 

57 

74 

8.510  6488 

819 

30 

94 

58 

70 

75 

844 

30 

98 

59 

66 

02 

869 

30 

6. 1902 

60 

8. 509  0162 

8.  510  5449 

1. 38894 

2.  3930 

6.  1905 

FACTOES  FOR  OOMPUTATIOS^  OF  GEODETIC  POSITIONS.        209 


Table  XXVII. — Factors  for  the  compulation  of  yeodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  44°. 


log.  A 

log.B 

log.C 

log.  D 

log.E 

Latitude. 

diff.  1"=— 0.07 

diff.  1"=— 0.21 

difl-.  l"=+0.42 

diff.  l"=+0.00 

diff.  l"=+0.06 

44  00 

8.  509  0162 

8.510  5449 

1.  38894 

2. 3930 

6. 1905 

1 

57 

36 

919 

30 

09 

2 

53 

23 

945 

30 

13 

3 

49 

01 

970 

30 

17 

4 

44 

8. 510  5388 

995 

30 

20 

05 

40 

75 

1. 39020 

31 

24 

6 

36 

62 

045 

31 

28 

7 

31 

49 

070 

31 

31 

8 

27 

36 

095 

31 

35 

9 

23 

23 

120 

31 

39 

10 

3. 509  0119 

8.  510  5311 

1. 39145 

2. 3931 

6. 1943 

11 

14 

07 

171 

31 

46 

12 

10 

8.  510  5295 

196 

fl 

50 

13 

06 

82 

221 

31 

54   • 

14 

02 

09 

246 

31 

58 

15 

8. 509  0097 

56 

271 

31 

61 

16 

93 

43 

296 

31 

6b 

17 

89 

30 

321 

32 

69 

18 

84 

18 

346 

32 

72 

19 

80 

05 

371 

32 

76 

20 

8.  509  0076 

9. 510  5192 

1.  39396 

2.  3932 

6. 1980 

21 

72 

79 

422 

32 

84 

22 

67 

66 

447 

32 

87 

23 

63 

53 

472 

32 

91 

24 

59 

40 

497 

52 

95 

25 

54 

28 

622 

32 

99 

26 

50 

15 

547 

32 

6.  2002 

27  . 

46 

02 

572 

32 

06 

28 

42 

8.  510  5089 

697 

32 

10 

29 

37 

76 

623 

32 

14 

30 

8.  509  0033 

8. 510  5063 

1.39648 

2.  3932 

6.  2017 

31 

29 

50 

673 

32 

21 

32 

24 

37 

698 

H2 

25 

-   33 

20 

25 

723 

33 

29 

34 

16 

12 

748 

33 

32 

35 

11 

8.510  4999 

773 

33 

36 

36 

07 

86 

79S 

33 

40 

37 

03 

73 

823 

33 

44 

38 

8.  508  9999 

60 

848 

33 

47 

39 

94 

47 

873 

33 

51 

40 

8.  508  9990 

8. 510  4935 

1.39898 

2.  3933  . 

6. 2055 

41 

86 

22 

924 

33 

59 

42 

81 

09 

949 

33 

62 

43 

77 

8.  510  4896 

974 

33 

66 

44 

73 

83 

999 

33 

70 

45 

69 

70 

1. 40024 

33 

74 

46 

64 

57 

049 

33 

77 

47 

60 

44 

074 

33 

81 

48 

56 

32 

099 

33 

85 

49 

51 

19 

124 

33 

89 

50 

8.  508  9947 

8. 510  4806 

1. 40149 

2. 3933 

6.2092 

51 

43 

8. 510  4793 

174 

33 

96 

52 

39 

80 

200 

33 

6.  2100 

53 

34 

67 

225 

33 

04 

54 

30 

54 

250 

33 

08 

55 

26 

41 

275 

33 

11 

56 

21 

29 

300 

33 

15 

57 

17 

16 

325 

33 

19 

58 

13 

03 

350 

33 

23 

59 

09 

8. 510  4690 

375 

33 

27 

60 

8. 508  9904 

8. 510  4677 

1.  40400 

2.  3933 

6.  2130 

-14 


210 


A  MANUAL  OF  TOPOGEAPHIO  METHODS. 


Table  XXVII.— Fne/ors  for  Ihe  c<>m}>ittalioii  of  geodetic  latitudes,  longitudes,  and  azimuths— Contimied. 

LATITUDE  -ISO. 


log.  A 
!.liff.l"=-0.07 


log.  B 
<litt'.l"  =  — 0.21 


log.  C 

iliir.l"  =  H-0.42 


log.  D 

dlff.l"=±0.00 


log.  E 
cliff.  1"  = +0.06 


FACTORS  FOR  COMPUTATION  OF  GEODETIC  POSITIONS.       211 


Table  XXVII.— Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  asmui^tg— Continued. 

LATITUDE  46°. 


log.  A 

log.B 

log.C 

log.D 

log.E 

Latitude 

(liff.  ]"=— 0.07 

diff.  1"=— 0.21 

diff.  1"=  +0.42 

difif.  1"=— 0.00 

diff.  1"= +0.06 

46  00 

8. 508  9647 

8. 510  3905 

1. 41906 

2.  3932 

6. 2359 

1 

43 

8. 510  3892 

931 

32 

63 

2 

38 

79 

957 

31 

67 

3 

34 

67 

982 

31 

71 

4 

30 

54 

1. 42007 

31 

75 

05 

25 

41 

032 

31 

79 

6 

21 

28 

057 

31 

82 

7 

17 

15 

082 

31 

86 

8 

13 

02 

107 

31 

90 

9 

08 

8.510  3739 

132 

31 

94 

10 

8.  508  9604 

8.510  3776 

1.42157 

2.3931 

6.  2398 

11 

00 

64 

183 

31 

6.  2402 

12 

8.  508  959S 

51 

208 

31 

06 

13 

91 

38 

233  • 

30 

09 

14 

87 

25 

268 

30 

13 

15 

83 

12 

283 

30 

17 

16 

78 

8. 510  3699 

308 

30 

21 

17 

74 

86 

333 

30 

25 

18 

70 

74 

358 

30 

29 

19 

65 

61 

.   384 

30 

33 

20 

8.  508  9561 

8.  510  3648 

1. 42409 

2.  3930 

6. 2436 

21 

57 

35 

434 

30 

40 

22 

53 

22 

459 

30 

44 

23 

'48 

09 

484 

29 

48 

24 

44 

8.  510  3596 

509 

29 

52 

25 

40 

84 

534 

29 

56 

26 

35 

71 

559 

29 

60 

27 

31 

58 

584 

29 

64 

28 

27 

45 

610 

29 

67 

29 

23 

32 

635 

39 

71 

30 

8.  508  9518 

8. 510  3519 

1.42660 

2. 3929 

6.  2475 

31 

14 

06 

685 

29 

79 

32 

10 

8.  510  3494 

710 

28 

83 

33 

05 

81 

735 

28 

87 

34 

01 

68 

760 

28 

91 

35 

8. 508  9497 

55 

786 

28 

95 

36 

93 

42 

811 

28 

99 

37 

88 

29 

836 

28 

8, 2502 

38 

84 

17 

861 

28 

06 

39 

80 

04 

886 

28 

10 

40 

8. 508  9475 

8. 510  3391 

1.  42911 

2.  3927 

6.  2514 

41 

71 

78 

936 

27 

18 

42 

67 

65 

961 

27 

22 

43 

63 

52 

987 

27 

26 

44 

58 

39 

1.  43012 

27 

30 

45 

54 

27 

037 

27 

34 

46 

50 

14 

062 

27 

38 

47 

45 

01 

087 

26 

41 

48 

41 

8.  510  3288 

112 

26 

45 

49 

37 

75 

137 

26 

49 

50 

8.508  9433 

8.  510  3262 

1.43163 

2.3926 

6. 2553 

51 

28 

49 

188 

26 

57 

52 

24 

37 

213 

26 

61 

53 

20 

24 

238 

26 

65 

54 

16 

11 

263 

25 

69 

55 

11 

8.510  3198 

288 

25 

73 

56 

07 

85 

314 

25 

77 

57 

03 

72 

339 

25 

81 

58 

8.508  9398 

60 

364 

25 

84 

59 

94 

47 

389 

25 

88 

60 

8.508  9390 

8.  510  3134 

1. 43414 

2. 3924 

6.  2592 

212 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXVII. — Factors  for  ike  oompulallon  of  geodetic  latitudes,  longitudes,  and  azimuths— Continued. 

LATITUDE  47°. 


log.  A 

log.B 

log.C 

log.D 

log.E 

Latitude. 

(liff.  l"=.-0.07 

<litt-.  1"=— 0.21 

diB'.  l"=+0.42 

dift'.  1"=— 0.00 

ditl'.  l"=+0.07 

47  00 

8.  508  9390 

8.510  3134 

1.43414 

2.  3924 

6.2592 

1 

86 

21 

439 

24 

96 

2 

81 

08 

465 

24 

6.  2600 

3 

77 

8.510  3095 

490 

24 

04 

4, 

73 

82 

515 

24 

08 

05 

08 

70 

540 

24 

12 

6 

64 

57 

565 

23 

16 

7 

00 

44 

590 

23 

20 

8 

56 

31 

615 

23 

24 

9 

51 

18 

641 

23 

28 

10 

8. 508  9347 

8. 510  3005 

1.43666 

2.  3923 

6.  2632 

11 

43 

8.  510  2993 

691 

23 

35 

12 

'   38 

SO   ■ 

716 

22 

39 

13 

34 

•  67 

741 

22 

43 

14 

30 

54 

760 

22 

47 

15 

26 

41 

792 

22 

51 

16 

21 

28 

817 

22 

55 

17 

17 

16 

842 

21 

59 

18 

13 

03 

867 

21 

63 

19 

09 

8. 510  2890 

892 

21 

67 

20 

8,  508  9304 

8.  510  2877 

1.43917 

2.  3921 

6.  2671 

21 

00 

64 

943 

21 

75 

22 

8. 508  9296 

51 

968 

20 

79 

23 

91 

39 

993 

20 

83 

24 

87 

26 

1.44018 

20 

87 

25 

83 

13 

043 

20 

91 

26 

79 

00 

069 

20 

95 

74 

8.  510  2787 

094 

19 

99 

28 

70 

74 

119. 

19 

6.  2702 

29 

66 

62 

144 

19 

00 

30 

8.508  9261 

8.  510  2749 

1.44169 

2.  3919 

6.2710 

31 

57 

36 

195 

19 

14 

32 

53 

23 

220 

18 

18 

33 

49 

10 

245 

18 

22 

34 

44 

8. 510  2698 

270 

18 

26 

35 

40 

85 

295 

18 

30 

36 

36 

72 

321 

18 

34 

37 

32 

59 

346 

17 

38 

38 

27 

46 

371 

17 

42 

39 

23 

33 

,396 

17 

a 

40 

8.  508  9219 

8.  510  2621 

1.44421 

2.  3917 

6.  2750 

41 

14 

08 

447 

16 

54 

42 

10 

8. 510  2595 

472 

16 

68 

43 

06 

82 

497 

16 

62 

44 

02 

69 

522 

16 

66 

45 

8.508  9197 

57 

547 

16 

70 

46 

93 

44 

573 

15 

74 

47 

89 

31 

598 

15 

78 

48 

84 

18 

623 

15 

82 

49 

80 

05 

648 

15 

86 

50 

8.  508  9176 

8.  510  2493 

1.  44673 

2.  3914 

6.  2790 

51 

72 

80 

699 

14 

94 

52 

67 

67 

724 

14 

98 

53 

63 

54 

749 

14 

6.  2802 

54 

59 

41 

774 

13 

06 

55 

55 

28 

800 

13 

10 

50 

50 

16 

825 

13 

14 

57 

46 

03 

850 

13 

18 

58 

42 

8. 510  2390 

875 

12 

22 

59 

38 

77 

900 

12 

26 

60 

8. 508  9133 

8. 510  2364 

1.44926 

2. 3912 

6. 2830 

FACTORS  FOE  COMPUTATION  OF  GEODETIC  POSITIONS.       213 

Table  XXVIl.^Factors  for  the  computation  of  geodetic  latitudes,  lougitttdes,  and  azimuths — Continued. 

LATITUDE  48°. 


loj;.  A 

log.  B 

log.  C 

log.  h 

log.  E 

Lalitiule. 

(liff.l"  =— 0.07 

cliflf.l"=— 0.21 

lUff.  l"=+0.42 

diflU"=— 0.00 

(lifif.  l"  =  +0.07 

48  00 

8. 508  91B3 

8.  510  2364 

1. 44926 

2. 3912 

6. 2830 

1 

29 

52 

951 

12 

34 

2 

25 

39 

976 

11 

38 

3 

20 

26 

1.45001 

11 

42 

i 

16 

13 

027' 

n 

46 

05 

12 

00 

052 

11 

50 

6 

08 

8.510  2288 

077 

10 

54 

7 

03 

75 

102 

10 

58 

8 

8.  5118  9099 

62 

128 

10 

62 

9 

95 

49 

153 

10 

66 

10 

8. 508  9091 

8. 510  2236 

■   1.45178 

2. 3909 

6.2870 

11 

86 

24 

203 

0!) 

74 

12 

82 

11 

229 

09 

78 

13 

78 

8. 510  2198 

254 

08 

82 

14 

74 

85 

279 

08 

86 

15 

69 

72 

304 

08 

90 

16 

65 

60 

330 

08 

94 

17 

61 

47 

355 

07 

98 

18 

57 

34 

380 

07 

6.2902 

19 

52 

21 

406 

07 

06 

20 

8. 508  9048 

8. 510  2108 

1.45431 

2.3907 

6.  2910 

21 

44 

8.510  2096 

456 

06 

14 

22 

39 

83 

481 

06 

18 

23 

35 

70 

507 

06 

22 

?i 

31 

57 

532 

05 

26 

25 

27 

45 

557 

05 

30 

26 

22 

32 

582 

05 

34 

27 

18 

19 

608 

05 

38 

28 

14 

06 

633 

04 

42 

29 

10 

8.  510  1993 

658 

04 

46 

30 

8. 508  9005 

8. 510  1981 

1.45683 

2.  3904 

6.  2950 

31 

01 

68 

709 

03 

54 

32 

8.  508  8997 

55 

734 

03 

58 

33 

93 

42 

759 

03 

62 

34 

88 

30 

785 

02 

«6 

35 

84 

17 

810 

02 

70 

36 

80 

04 

835 

02 

74 

37 

76 

8. 510  1891 

861 

02 

78 

38 

71 

78 

886 

01 

82 

39 

67 

66 

911 

01 

86 

40 

8.  508  896i 

8.  510  1853 

1. 45937 

2.  3901 

6.  2990 

41 

59 

40 

962 

00 

94 

42 

54 

27 

987 

00 

98 

43 

50 

15 

1.46012 

00 

6.  3002 

44 

46 

02 

038 

2. 3899 

06 

45 

41 

8. 510  1789 

063 

99 

10 

46 

37 

76 

088 

99 

15 

47 

33 

64 

114 

98 

19 

48 

29 

51 

139 

98 

23 

49 

24 

38 

164 

98 

27 

50 

8. 508  8920 

8.510  1725 

1.46190 

2.  3897 

6. 3031 

51 

16 

13 

215 

97 

35 

52 

12 

00 

240 

97 

39 

53 

08 

8.  510  1687 

266 

96 

43 

54 

03 

74 

291 

96 

47 

55 

8. 508  8899 

62 

316 

96 

51 

56 

95 

49 

342 

95 

55 

57 

90 

36 

367 

95 

59 

58 

86 

23 

392 

95 

63 

59 

82 

10 

418 

94 

67 

60 

8.  508  8878 

8.510  1598 

1.46443 

2. 3894 

6. 3071 

214 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXVU.— Factors  for  the  comimtation  of  geodetic  latitudes,  longitudes,  and  azimuths— ContinneA. 

LATITUDE  49°. 


log.  A 
diff.l"=— 0.07 


log.B 
diff.l"=— 0.21 


log.  0 
fliff.l"=:+0.42 


log.D 
diff.  1"=— 0. 01 


1.  508  8708 
04 
00 

1.508  8695 


FACTOES  FOR  COMPUTATION  OF  GEODETIC  POSITIONS.        215 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 

LATITUDE  30°. 


Latitude. 

log.  A 

log.  B 

log.  C 

log.  D 

log.  E 

dift'.  1"=— 0.07 

dlfif.  1"=_0.21 

diff.  1"=X0.43 

diff.  1"=— 0.01 

diir.  l"=+0.07 

60  0 

8.  608  8623 

8.  510  0835 

L 47968 

2. 3871 

6. 3318 

1 

19 

22 

993 

70 

22 

2 

15 

09 

1.  48019 

70 

26 

3 

11 

8.  510  0797 

044 

70 

30 

4 

06 

84 

070 

69 

34 

05 

02 

71 

096 

69 

39 

6 

8. 508  8598 

.59 

121 

68 

43 

7 

94 

46 

146 

68 

47 

8 

90 

33 

172 

67 

51 

9 

85 

21 

197 

67 

55 

10 

8.  508  8581 

8.  510  0708 

1.48223 

2.  3866 

6.  3369 

11 

77 

8. 510  0695 

248 

66 

63 

12 

73 

83 

274 

66 

68 

13 

68 

70 

299 

66 

72 

14 

64 

57 

325 

65 

76 

15 

60 

45 

350 

64 

80 

16 

56 

32 

376 

64 

84 

17 

52 

19 

401 

63 

88 

18 

47 

07 

427 

63 

93 

19 

43 

8. 510  0594 

452 

62 

97 

20 

8.  508  8539 

8. 510  0581 

1. 43478 

2.3862  . 

6.  3401 

21 

35 

69 

504 

61 

06 

22 

30 

56 

529 

61 

09 

23 

26 

43 

655 

60 

14 

24 

22 

31 

580 

60 

18 

25 

18 

18 

606 

60 

22 

26 

14 

06 

631 

59 

26 

27 

09 

8. 510  0493 

657 

69 

30 

28 

05 

80 

682 

58 

34 

'  29 

01 

67 

708 

58 

39 

30 

8.  508  8497 

8. 510  0455 

1.48734 

2. 3857 

6.  3443 

31 

93 

42 

759 

67 

47 

32 

88 

29 

785 

56 

61 

33 

84 

17 

810 

66 

55 

34 

80 

04 

836 

65 

60 

35 

76 

8.  510  0392 

861 

55 

64 

36 

71 

79 

887 

54 

68 

87 

07 

66 

913 

54 

72 

38 

63 

54 

938 

53 

76 

39 

59 

41 

964 

53 

81 

40 

8. 508  8455 

8. 510  0328 

1.  48989 

2.  3852 

6.  3485 

41 

50 

16 

1.49015 

52 

89 

42 

46 

03 

041 

51 

93 

43 

42 

8.5W0291 

066 

51 

97 

44 

38 

78 

092 

50 

6. 3502 

45 

34 

65 

117 

50 

06 

46 

29 

53 

143 

49 

10 

47 

26 

40 

169 

49 

14 

48 

21 

27 

194 

48 

18 

49 

17 

16 

220 

48 

23 

50 

8.  508  8413 

8.  510  0202 

1. 49246 

2.  3847 

6.  3527 

51 

08 

8. 510  0190 

271 

47 

31 

52 

04 

77 

297 

46 

35 

63 

00 

64 

322 

46 

40 

54 

8.  508  8396 

62 

348 

45 

44 

55 

92 

39 

374 

45 

48 

56 

87 

27 

399 

44 

52 

57 

83 

14 

425 

44 

56 

58 

79 

01 

451 

43 

61 

59 

75 

8.  510  0089 

476 

43 

65 

60 

8. 508  8371 

8.510  0076 

1.49502 

2. 3842 

6.3569 

216 


A  MAifUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXVII. — Factors  for  the  computation  of  geodetic  latitudes,  longitudes,  and  azimuths — Continued. 
COEEEOTIOXS  TO  LONGITUDE  FOR  DIFFERENCE  IN  ARC  AND  SINE. 


Log.  K 
(-) 

Log.  difference. 

Log.  d  M 
(+) 

Log.  K 

(-) 

Log.  difference. 

Log.dM 

(+) 

Log.K 

Log.  difference. 

Log.dM 
(+) 

3.876 

0.000  0001 

2.385 

4.813 

0. 000  0075 

3.322 

5.114 

0. 000  0300 

3.623 

4.026 

02 

2.  535 

4.825 

080 

3.334 

5.120 

309 

3.629 

4.114 

03 

2.623 

4.834 

084 

3.343 

5.126 

318 

3.635 

4.177 

04 

2.686 

4.849 

089 

3.358 

5.132 

327 

3.641 

4.225 

05 

2.734 

4.860 

094 

3.369 

5.138 

336 

3.647 

4.265 

06 

2.774 

4.871 

098 

3.380 

5.144 

345 

3.653 

4.298 

07 

2.807 

4.882 

103 

3.391 

5.150 

354 

3.659 

4.327 

08 

2.836 

4.892 

108 

3.401 

5.156 

364 

3.665 

4.353 

09 

2.862 

4.903 

114 

3.412 

5.161 

373 

3.670 

4.376 

10 

2.885 

4.913 

119 

3.422 

5.167 

383 

3.676 

4.396 

11 

2.905 

4.922 

124 

3.431 

5.172 

392 

3.681 

4.415 

12 

2.924 

4.932 

130 

3.441 

5.178 

402 

3.687 

4.433 

.  13 

2.942 

4.941 

136 

3.450 

5.183 

412 

3.692 

4.449 

14 

2.958 

4.950 

142 

3.459 

5.188 

422 

3.697 

4.464 

15 

2.973 

4.959 

147 

3.463 

5.193 

433 

3.702 

4.478 

16 

2.987 

4.968 

153 

'3.477 

5.199 

443 

3.708 

4.491 

17 

3.000 

4.976 

160 

3.485 

5.204 

453 

3.713 

4.503 

18 

3.012 

4.985 

166 

3.494 

5.209 

464 

3.718 

4.526 

20 

3.035 

4.993 

172 

3.502 

5.214 

474 

3.723 

4.548 

23 

3.057 

5.002 

179 

3.511 

5.219 

486 

3.728 

4.570 

25 

3.079 

5.010 

186 

3.519 

5.223 

497 

3.732 

4.591 

27 

3.100 

5.  017 

192 

3.526 

5.  228 

508 

3.737 

4.612 

30 

3.121 

5.025 

199 

3.534 

5.233 

519 

3.742 

4.631 

33 

3.140 

1   5.033 

206 

3.542 

5.238 

530 

3.747 

4.649 

36 

3.158 

5.040 

213 

3.549 

5.242 

541 

3.761 

4.667 

39 

3.176 

5.047 

221 

3.  556 

5.247 

553 

3.756 

4.684 

.42 

3.193 

5.054 

228 

3.563 

5.251 

565 

3.760 

4.701 

45 

3.210 

5.062 

236 

3.571 

5.256 

577 

3.765 

4.716 

48 

3.225 

5.068 

243 

3.577 

5.260 

588 

3.769 

4.732 

52 

3.241 

5.075 

251 

3.584 

5.265 

600 

3.774 

4.746 

56 

3.255 

5.  082 

259 

3.591 

5.269 

613 

3.778 

4.761 

59 

3.  270 

5.088 

267 

3.597 

5.273 

625 

3.782 

4.774 

63 

3.283 

5.  095 

275 

3.604 

5.278 

637 

3.787 

4.788 

67 

3.297 

5. 102 

284 

3.611 

5.282 

650 

3.791 

4.  801 

71 

3.  310 

5. 108 

292 

3.617 

5.286 

663 

3.795 

FACTOES  FOE  EEDUGTION  OF  TEANSIT  OBSEEVATIONS.       217 


Table  XXVIII. — Factors  for  reduction  of  transit  ohservations. 

[Extracted  from  Appendix  14,  U.  S.  Coast  and  Geodetic  Survey  Report  for  1880.] 

To  find  A  enter  left-hand  column  -with,  tlie  zenith  ^^distance;  its  intersection  "with  declination  column  gives  azimuth 
factor. 

To  find  B  enter  ri^ht-liand  column  with  tlie  zenith  distance ;  its  intersection  with  declination  column  gives  level  factor. 
C  is  given  on  laat  line  of  each  section  of  the  table. 


Azimuth,  factor  A=sin,  ^  sec.  6.    Star's  declination  : 


Inclination  factor  B^cos  ^  sec.  A. 


i 

0° 

10° 

15^ 

20° 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

36° 

38° 

40°  41° 

42° 

430 

'44° 

45° 

46° 

47° 

48° 

490 

50° 

i 

1° 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.02 

.03 

.02 

.02 

i 
.02:  .02 

.02 

.02 

.02 

.02 

.02 

.03 

.03 

.03 

.03 

89° 

2 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.04 

.05 

.06 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

.05 

88 

3 

.05 

.05 

.05 

.06 

.06 

.06 

.00 

.06 

.06 

.06 

.06 

.00 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

87 

4 

.07 

.07 

.07 

.07 

.08 

.08 

.08 

.08 

.08 

.08 

.08 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

86 

5 

.09 

.09 

.09 

.09 

.09 

.10 

.10 

.10 

.10 

.10 

.10 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.;2 

.13 

.13 

.13 

.13 

.13 

85 

6 

.11 

.11 

.111 

.11 

.11 

.11 

.12 

:i4 

.12 

.12 

.13 

.13 

.13 

.14 

.14 

.14 

.14 

.16 

.15 

.15 

.15 

.16 

.16 

.16 

84 

7 

.12 

.12 

.131 

.13 

.13 

.13 

.11 

.14 

.14 

.15 

.15 

.15 

.16 

.16 

.16 

.17 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

83 

8 

.14 

.14 

.14 

.15 

.15 

.15 

.16 

.16 

.16 

.16 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.19 

.20 

.20 

.20 

.21 

.21 

.22 

82 

9 

.16 

.16 

.16 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.24 

.24 

81 

lo- 

.17 

.18 

.18 

.19 

.19 

.19 

.19 

.20 

.20 

.21 

.21 

.21 

.22 

.23 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.26 

.26 

.27 

SO 

ll 

.19 

.19 

.20' 

.20 

.21 

.21 

.21 

.22 

.22 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.26 

.27 

.27 

.28 

.28 

.28 

.29 

.30 

79 

12 

.21 

.21 

.22 

.22 

.  22 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.26 

.27 

.27 

,28 

.28 

.29 

.29 

.30 

.30 

.31 

.32 

.32 

78- 

13 

.  22 

.23 

.23' 

.24 

.24 

.25 

.25 

.26 

.26 

.27 

.27 

.28 

.29 

.29 

.30 

.30 

.31 

.31 

.32 

.32 

.33 

.34 

.34 

.35 

77 

14 

.2+ 

.25 

.251 

.26 

.26 

.27 

.27 

.27 

.28 

.29 

.29 

.30 

.31 

.32 

.32 

.33 

.33 

.34 

.34 

.35 

..36 

.36 

.37 

.38 

76 

15 

.26 

.26 

.271 

.28 

.28 

.28 

.29 

.29 

.30 

.31 

.31 

.32 

.33 

.34 

.34 

.36 

.35 

.36 

.37 

.37 

.38 

.39 

.39 

.40 

76 

10 

.28 

.28 

.29, 

.29 

.30 

.30 

.31 

.31 

.32 

.33 

.33 

.34 

.35 

.36 

.37 

.37 

.38 

.38 

.39 

.40 

.40 

.41 

.42 

.43 

.74 

17 

.29 

.30 

.30 

.31 

.31 

.32 

.33 

.33 

.34 

.34 

.35 

.36 

.87 

.38 

.39 

.39 

.40 

.41 

.41 

'.42 

.43 

.44 

.45 

.45 

73 

IS 

.3; 

.31 

.32 

.33 

.33 

.3a 

.34 

.35 

.36 

.36 

.37 

.38 

.39 

.40 

.41 

.42 

.42 

.43 

.44 

.44 

.45 

.46 

.47 

.48 

72 

19 

.33 

.33 

.34 

.35 

.35 

.36 

.36 

.37 

.38 

.38 

.39 

.40 

.41 

.42 

.43 

.44 

.45 

.45 

.40 

.47 

.48 

.49 

.50 

.51 

71 

20 

.34 

.35 

.35 

.36 

.37 

.37 

.38 

.39 

.40 

.40 

.41 

.42 

.43 

.45 

.45 

.46 

.47 

.48 

.48 

.49 

.50 

.51 

.62 

.53 

JO 

21 

.36 

.36 

.37 

.38 

.39 

.39 

.40 

.41 

.41 

.42'  .43 

.44 

.45 

.47 

.47 

.48 

.49 

.50 

.51 

.52 

..52 

.54 

.55 

.66 

69 

22 

.37 

.38 

.39 

.40 

.40 

.41 

.42 

.42 

.43 

.44'  .45 

.46 

.48 

.49 

.50 

.50 

.61 

..52 

.  53 

.54 

.55 

.56 

.67 

.58 

68 

23 

.39 

.40 

.41 

.42 

.42 

.43 

.44 

.44 

.45 

.46|  .47 

.48 

.50 

.51 

.52 

.53 

.53 

.51 

.  5ii 

.56 

.  57 

.58 

.60 

.61 

67 

24 

.41 

.41 

.42 

.43 

.44 

.45 

.45 

.46 

.47 

.48 

.49 

.60 

.52 

.53 

.54 

.55'  .56 

.67 

.58 

.59 

.00 

.61 

.62 

.63 

66 

25 

.42 

.43 

.44 

.45 

.46 

.46 

.47 

.48 

.49 

.60 

.51 

.52 

.64 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66 

66 

26 

.44 

.45 

.45 

.47 

.47 

.48 

.49 

.50 

.51 

.62 

.53 

.54 

.56 

.67 

.58 

.69 

.60 

.61 

.62 

.63 

.64 

.65 

.67 

.68 

64 

27 

.45 

.46 

.47 

.48 

.49 

.50 

.51 

.51 

.62 

.54 

.55 

.66 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66 

.67 

.68 

.69 

.71 

63 

28 

.47 

.48 

.49 

.50 

.51 

.51 

.52 

.53 

.54 

.56 

.67 

.58 

.60 

.61 

.62 

.63 

.64 

.65 

.66 

.68 

.69 

.70 

.72 

.73 

02 

29 

.48 

.49 

.50 

.52 

.52 

.53 

.54 

.55 

.56 

.57 

.58 

.60 

.61 

.63 

.64 

.66 

.66 

.67 

.69 

.70 

.71 

.72 

.74 

.75 

61 

30 

.50 

.51 

.62 

.53 

.64 

.55 

.56 

.57 

.68 

.59 

.60 

.62 

.63 

.65 

.66 

.67 

.68 

.69 

.71 

.72 

.73 

.75 

.76 

.78 

(iO 

31 

.52 

.52 

.53 

.55 

.56 

.56 

.67 

.58 

.59 

.61 

.62 

.64 

.65 

.67 

.68 

.69 

.70 

.72 

.73 

.74 

.75 

.77 

.78 

.80 

59 

32 

.63 

.54 

.56 

..56 

.57 

.58 

.69 

.60 

.61 

.63 

.64 

.66 

.67 

.69 

.70 

.71 

.72 

.74 

.75 

.76 

.78 

.79 

.81 

.82 

58 

33 

.54 

.55 

56 

.58 

.69 

.60 

.61 

.62 

.63 

.64 

.66 

.67 

.69 

.71 

.72 

.73 

.74 

.76 

.77 

.78 

.80 

.81 

.83 

.85 

57 

34 

.56 

.57 

.58 

.159 

.60 

.61 

.62 

.63 

.65 

.66 

.67 

.69 

.71 

.73 

.74 

.75 

.70 

.78 

.79 

.80 

.82 

.84 

.85 

.87 

66 

35 

.57 

.58 

.59 

.61 

.62 

.63 

.64 

.65 

.66 

.68 

.69 

.7) 

.73 

.75 

.76 

.77 

.78 

.80 

.81 

.83 

.84 

.86 

.87 

.89 

55 

36 

.59 

.60 

,61 

.63 

.63 

.64 

.65 

.67 

.68 

.69 

.71 

.73 

.75 

.77 

.78 

.79 

.80 

.82 

.83 

.85 

.86 

.88 

.90 

.91 

54 

37 

.60  .61 

.62 

.64 

.65 

.65 

.67 

.68 

.70 

.71 

.73 

.74 

.76 

.79 

.80 

.81 

.82 

.84 

.85 

.87 

.88 

.90 

.92 

.94 

63 

38 

.62  .63 

.64 

.66 

.66 

.67 

.69 

.70 

.71 

.73 

.74 

.76 

.78 

.80 

.82 

.83 

.84 

.86 

.87 

.89 

.90 

.92 

.94 

.96 

52 

39 

.63  .64 

.65 

.67 

.68 

.69 

.70 

.71 

.73 

.74 

.76 

.78 

.80 

.82 

.83 

.85 

.86 

.87 

.89 

.91 

.92 

.94 

.96 

.98 

51 

10 

.64 

.65 

.66 

.68 

.69 

.70 

.72 

.73 

.74 

.76 

..77 

.79 

.82 

.84 

.85 

.86 

.88 

.89 

.91 

.93 

.94 

.96 

.98 

1.00 

50 

41 

.66 

.67 

.68 

.70 

.71 

.72 

.73 

.74 

.76 

.77 

.79 

.81 

.83 

.86 

.87 

.88 

.90 

.91 

.93 

.94 

.90 

.98 

1.00 

1.02 

49 

42 

.67 

.68 

.69 

.71 

.72 

.73 

.74 

.76 

.77 

.79 

.81 

.83 

.85 

.87 

.89 

.90 

.91 

.93 

.95 

.96 

.9S 

1.00 

1.  02 

1.04 

48 

43 

.68 

.69 

.71 

.73 

.74 

.75 

.76 

.77 

.79 

.80 

.82 

.84 

.86 

.89 

.90 

.92 

.93 

.95 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

47 

44 

.69 

.71 

.72 

.74 

.75 

.76 

.77 

.79 

.80 

.82 

.84 

.86 

.89 

.90 

.92 

.93 

.96 

.96 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

46 

45 

.71 

.72 

.73 

.75 

.76 

.77 

.79 

.80 

.82 

.83 

.85 

.87 

.90 

.92 

.94 

.95 

.97 

.98 

1.00 

1.02 

1.04 

1.06 

1.08 

1.10 

46 

46 

.72 

.73 

.74' 

.77 

.78 

.79 

.80 

.82 

.83 

.85 

.87 

.89 

.91 

.94 

.95 

.971  .98 

1.00 

1.  02 

1.04 

1.06 

1.07 

1.10 

1.12 

44 

47 

.73 

.74 

.76 

.78 

.79 

.80 

.81 

.83 

.84 

.86 

,88 

.90 

.93 

.95 

.97 

.98,1.00 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

1.14 

43 

48 

.74 

.76 

.77 

.79 

.80 

.81 

.83 

.84 

.86 

.88 

.90 

.92 

.94 

.97 

.98  1.00:1.  02 

1.03 

1.  05 

1.07 

1.09 

1.11 

1.13 

1.16 

42 

49 

.75 

.77 

.78 

.80 

.81 

.83 

.84 

.86 

.87 

.89 

.91 

.93 

.96 

.99 

1.001.02;!.  03 

1.05 

1.  07 

1.09 

1.11 

1.13 

1.15 

1.17 

41 

60 

.77 

.78 

.79 

.82 

.83 

.84 

.85 

.87 

.89 

.90 

.92 

.95 

.97 

1.00 

1.011.031.05 

1.06 

1.  08 

1.10 

1.12 

1.14 

1.17 

1.19 

40 

51 

.78 

.79 

.80 

.83 

.84 

.85 

.87 

.88 

.90 

.92 

.94 

,96 

.99 

1.01 

1.031.051.06 

1.  08 

1.  10 

1.12 

1.14 

1.16 

1.18 

1.21 

39 

52 

.79 

.80 

.82 

.84 

.85 

.86 

.88 

.89 

.91 

.93 

.95 

.97 

1.00 

1.03 

1.041.0611.08 

1.10 

1.  11 

1.13 

1.15 

1.18 

1.20 

1.23 

33 

53 

.80 

.81 

.83 

.85 

.86 

.87 

.89 

.91 

.92 

.94 

.96 

.99 

1.01 

1.04 

1.061.071.09 

1.  11 

1.  13 

1.15 

1.17 

1.19 

1.  22 

1.24 

37 

54 

.81 

.82 

.84 

.86  .87 

.89 

.90 

.92 

.93 

.95 

.98 

1.00 

1.03 

1.06 

1.07  1.091.11 

1.  12 

1.  14 

1.16 

1.11 

1.21 

1.23 

1.26 

36 

55 

.82 

.83 

.85 

.87,  .88 

.90 

.91 

.93 

.95 

.97 

.99 

1.01 

1.01 

1.0- 

1.  08  1. 10  1. 12 

1.14 

1.16 

1.18 

1.20 

1.22 

1.25 

.1.27 

35 

56 

.83 

.84 

.86 

.88  .89 

.91 

.92 

.94 

.06 

.98 

1.00 

1.02 

1.05 

1.08 

1. 10  1. 12  1. 13 

1.16 

1.17 

1.19 

1.22' 1.24 

1.26 

1.29 

34 

57 

.84 

.85 

.87 

.89  .90 

.92 

.93 

.95 

.97 

.991.01 

1.04 

1.06 

1.09 

1.  11 1. 13  1.  15 

1.17 

1.19 

1.21 

1.23  1.25 

1.28 

1.31 

33 

58 

.85 

.86 

.88 

.90  .91 

.93 

.94 

.96 

.98 

1.  00  1.  02 

1.05 

1.08 

1.  11 

1. 12  1. 14  1. 16 

1.18 

1.20 

1.22 

1.24  1.27 

1.29 

1.32 

32 

59 

.86 

.87 

.89 

.91  .92 

.94 

.95 

.97 

.99 

1.011.03 

1-.06 

1.  09 

1.12 

1. 14  1.  15  1. 17 

1.19 

1.21 

1.23 

1.  26  1.  28 

1.31 

1.33 

31 

60 

.87 

.88 

.90 

.92  .93 

.96 

.96 

.98 

1.00 

1.  02  1.  04 

1.07 

1.10 

1.13 

1.161.1711.18 

1.20 

1.22 

1.25 

1.27  1.29 

1.32 

1.35 

SO 

218 


A  MA]!fUAL  OP  TOPOGRAPHIC  METHODS. 


Table  XXVIII. — Factors  for  reduction  of  transit  oiservations — Continued. 
Azimuth  factor  A:=sia  C  sec.  6.    Star's  declination  +  i.    Inclination  factor  B=coa  ^  sec.  6. 


0°!  10°   15°    20°   22°  ■  24°i  26°   28°    30°   32°   34°  36°  88°    40°  41°   42°   43°   44°    45°   46°   47° '  48°     49°     50° 


.%    .97    .9!)  11.01,1.  03 


.  98  1.  00 
.98]  .991.01 
.98  1.110  1.02 
.  99I1.  01 1.  03 


1.  02  1.  04 
1.0311.05 
I.O1I1.O6 
1.05,1.07 


.95  1  .97'  .991. 
.95/  .98  .9'.!  1. 
.96l  .9!)  l.UO  I. 
.97  .99  L.  Ill  1. 
.971  l.OOjl.  Olll. 

.98,i.01il.02'l. 
.98  1.01il.0:l|l. 
.99  1.02'l.03!l.i 
.OOjl.  02  1.041. 

.ooii.oaii.w'i. 


1.05,1.08ll.H:i.  141. 16 
1.06  1.09:1.12  1.151.17 
1.  07JHO  1.13,11.16  1.18 
1.  OS  1.  111.  14  1. 17  1. 19 
1.  09  1.  I3I1.  15  1.18,1.20 


05  1.  07 

06  1,  08 

06  1.  08 

07  1.  09 

08  1.  0!) 


1.091. 
1. 10  1. 
1. 10  1. 
1. 11 1. 
1. 12  1. 


.OSl. 
.  09  1. 

.ooli. 
.  10 1. : 
.111.: 

121.; 
12 1. : 
131.: 
13 1. : 
141.: 


.  13 1. 16 

.141.  17 
.  15  1.  1.^ 
.15  1.  18 
.161. 10; 

.17:1.20' 
.171.31 
.  18  1.  21I 
.191.22 
.191.23;' 


M9'I.2I  1.  23  1.25,1, 


1.2011.22 
1.  21 1.  23 
1.22  1.24 
1.23J1.  25 
1.241.  26  1.28 


1.  25  1. 
1.26,1. 
1.  27  1. 
1.  27  1. 
1.  28  1. 


.29  1. 
.  30  1. 
.811. 
.  31  1. 
.321. 


1.26 
1.27 
1.28 
1.20 
1.30 

29  1.  32 

30  1 

31  1.  33 

32  1.  34 

33  1.  35 
i 

341.36 

34  1.  37 

35  1.  38 

36  1.  38 

37  1.  39 


.  97    .  99  I.  00  1.  0;l  1.  05  1.  06  1.  08  1.  HI  I.  12  1.  14  1.  IT  1.  20  1.  2:1  1.  27  I.  29  1.  31 

.97    .9:11.01  1.04  1.11,1  l.iiT  l.llil.  10  il.  1:5  1.  IS  1.17  1.20  1.24  1.27  1.29  1.  31 

.98    .OOU.III  1,04  1.0.'.  1.07  1.110  1.  11   1.13  1.151.  IS  1.21  1.24  1.28  1.30  1.32 

.  98  1.  00  1.  02  1.  04  1.  00  1.  08  1.  0;l  1. 11   1. 13  1. 16  1.  IS  1.  21 1.  25  1.  281.  30  1.  :f2 


1.331.35  1.371.40 
1.331.35  1.381.40 
1.341.  36  1.381.41 
1.34  1.  36  1.391.41 


i  1.001.  02  1.05  1.06,1.  OSl.  10  1.12  11.14  1.16  1.19  1.221.25,  1.  29'1.  30  1.  331.351.  37  1.391.42 

!    ■ '      I      !■      :  i      '       i      '  '':■■!  ■      I       I 

81  '  .991.0Qtl.03'  1.051.  07|1.  08  1.101.  12  ,1. 14  1. 17  1. 19  1.  22  1.  25  1.  39  1.  31  1.  33  1.  351.  37  1.401.42 

82  I  .991.01;!.  03,  1.  05  1.  07il.  03  1. 10  1. 12  1.  Ul- 17  1. 19  1.  23  1.  26  ,1.29  1.  31 1.  33  1.  35  1.  38  1.401.43 

83  1  .  99  1.  Olll.  03'  1.  06  1. 07:1.  09  1. 10  1. 12  1. 15  1. 17  1.  20,1.  23  1.  26  1.  30  1.  32  1.  34  1.  36  1,  38  1.  40  1.  43 

84  I  .  9911.  Olll.  031:1.  06il.  07ll.  09  1. 11  1. 13  il.  15  1. 17  1.  20il,  33 1.  26  1.  30|l.  32  1.  34|1.  36  1.  38  1.  41 1. 43 

85  1. 00  1.  01 1.  03  1.  06  1.  07  1.  09  1. 11 1. 13  11.  15  1. 17  1.  20  1.  33  1.  26  1.  3oll.  3211.  34  1.  36  1.  38  1.  4lll.  43 


1.28  1.31 
1.  29  1,  32 

1.31  1.33 

1.32  1.34 
1.  33  1. 35 

1.34  1.371 

1.35  1.38 
1.30!  1. 

1.  37,  1.  40 

1.38  1.40 

1.39  1.41 
1.  39  1.  42 
1.40;i.43 
1.41  1.44 
1.  42  1.  44 

1.42' 1.45 
1.43;  1.46 

1.43  1.46 

1.44  1.4'i 
1.44  1.47 


1.49 
1.49 
1.50 

1.51 
1.52 
1.53 
1.53 

1.50  1.53 

1.51  1.54 
1.51;  1.54 
1.51'  1.54 

1.52  1.55 

1.53  1,55 


88  1. 00  1.  01.1.  03'  1.  06  1.  08  1.  09,1. 111.13  1.15  1. 18  1.  20 

89  1. 00  1. 02  1.  04' 1.  06  1.  08  1. 09: 1. 11  1. 13  1. 15  1. 18,1.  21 

90  1.  ODJl.  02,1.  04  1.  06,1.  08|l.  09  1.  llll.  13  1. 15|l.  18,1. 21 


1.231.27  1.30  1.321.  3  I  1.37  1.30  1.41  1,44  1.40  1.49'  1.52  1.55 
1.241.27  1.31  I.:i2,l.:i5,l. :i71. 39,  1.411.  44,1. 47  1.49  1.52  1.56 
1. 24:1.  27|il.  31,1.  32  1.  35,1.  37,1.  39  ,1.  41|1. 44  1.  47  1.  49,  1.  52;  1.  56| 


r-r-T-r 


FACTOES  FOE  REDUCTION  OF  TRANSIT  OBSEUVATIONS.       219 


Table  XXVIII. — Factors  for  reduction  of  transit  observations — Continued. 
Azimath  factor  A  =  sin  ^  sec.  6.     Star's  declination  ±  5.    Inclination  factor  B  =  cos  i  sec.  i. 


i 

51° 
.03 

52° 
.03 

53° 
.03 

54° 
.03 

55° 
.03 

56° 
.03 

57° 

58° 

59° 
.03 

60° 
.03 

60J° 
.01 

61° 
.04 

6U° 
.04 

62° 
.04 

621° 
.04 

63° 
.04 

63i° 

64° 

64i° 
.04 

65° 
.04 

65J 
.04 

66° 
.04 

66i° 
.04 

67° 
.04 

i 

1° 

1 
.  03  .  03 

.04  .04 

89° 

2 

.06 

.06 

.06 

.06 

.06 

.06 

.06  .07 

.07 

.07 

.07 

.07 

.07 

.07 

.08 

.08 

.08'  .08 

.08 

.08 

.08 

.09 

.09 

.09 

88 

3 

.08 

.08 

.09 

.09 

.09 

.08 

.  10|  .  10 

.10 

.10 

.11 

.11 

.11 

.11 

.11 

.12 

.13'  .13 

.12 

.13 

.13 

.13 

.13 

.13 

87 

4 

.11 

.11 

.12 

.12 

.12 

.12 

.13 

.13 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

.15 

.16,  .16 

.16 

.17 

.17 

.17 

.18 

.18 

86 

5 

.14 

.14 

.14 

.15 

.15 

.16 

.16 

.16 

.17 

.17 

.18 

.18 

.18 

.19 

.19 

.19 

.19 

.30 

.20 

.21 

.21 

.31 

.22 

.22 

85 

6 

.17 

.17 

.17 

.18 

.18 

.19 

.19 

.20 

.20 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.26 

.27 

84 

7 

.19 

.20 

.20 

.21 

.21 

.22 

.22 

.23 

.24 

.24 

.25 

.  35 

.26 

.26 

.26 

.27 

.27 

.38 

.28 

.29 

.29 

.30 

.31 

.31 

83 

8 

.22 

.23 

.23 

.24 

.24 

.25 

.26 

.26 

.27 

.28 

.28 

.39 

.29 

.30 

.30 

.31 

.31 

.32 

.32 

.33 

.34 

.34 

.35 

.36 

82 

9 

.25 

.25 

.26 

.26 

.27 

.28 

.29 

.29 

.30 

.31 

.32 

.32 

.33 

.33 

.34 

.35 

.35 

.36 

.36 

.37 

.38 

.39 

.39 

.40 

81 

10 

.28 

.28 

.29 

.30 

.30 

.31 

.32 

.33 

.34 

.35 

.35 

.36 

.36 

.37 

.38 

.38 

.39 

.40 

.40 

.41 

.42 

.43 

.43 

.44 

80 

11 

.30 

.31 

.32 

.32 

.33 

.34 

.35 

.36 

.37 

.38 

.39 

.39 

.40 

.41 

.41 

.43 

.43 

.44 

.44 

.45 

.46 

.47 

.48 

.49 

77 

12 

.33 

..34 

.35 

.35 

.36 

.37 

.38 

.39 

.40 

.42 

.42 

.43 

.44 

.44 

.45 

.46 

.47 

.47 

.48 

.49 

.50 

.51 

.53 

78 

13 

.36 

.36 

.37 

.38 

.39 

.40 

.41 

.42 

.44 

.45 

.46 

.46 

.47 

.48 

.49 

.50 

.50 

.51 

.52 

.53 

.54 

.55 

!56 

.58 

77 

14 

.38 

.39 

.40 

.41 

.42 

.43 

.44 

.46 

.47 

.48 

.49 

.50 

.51 

.52 

.52 

.53 

.54 

55 

.56 

.57 

.58 

.59 

.61 

.62 

76 

15 

.41 

.42 

.43 

.44 

.45 

.46 

.48 

.49 

.50 

.52 

.53 

.53 

.54 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.64 

.65 

.66 

75 

16 

.44 

.45 

.46 

.47 

.48 

.49 

.51 

.52 

.54 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

.66 

.68 

.69 

.71 

74 

17 

.46 

.47 

.49 

.50 

.51 

.52 

.54 

.55 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66!  .67 

.68 

.69 

.70 

.72 

.73 

.75 

73 

18 

.49 

.50 

.51 

.53 

.54 

.55 

.57 

.58 

.60 

.62 

.63 

.04 

.65 

.66 

.67 

.68 

.69  .70 

.72 

.73 

.74 

.76 

.77 

.79 

72 

19 

.52 

.53 

.54 

.5.'> 

.57 

.58 

.60 

.61 

.63 

.65 

.66!  .67 

.68 

.69 

.70 

.72 

.73 

.74 

.76 

.77 

.78 

.80 

.82 

.83 

71 

20 

.54 

.50 

.57 

.58 

.60 

.61 

.63 

.04 

.66 

.68 

.69 

.70 

.72 

.73 

.74 

.75 

.77 

.79 

.79 

.81 

.83 

.84 

.86 

.88 

70 

21 

.57 

.58 

.59 

.61 

.62 

.64 

.66 

.68 

.70 

.72 

.73 

.74 

.75 

.76 

.78 

.79 

.80 

.83 

.83 

.85 

.86 

.88 

.90 

.92 

69 

22 

.60 

.61 

.62 

.64 

.65 

.67 

.69 

.71 

.73 

.75 

.76 

.77 

.78 

.80 

.81 

.82 

.84 

.85 

.87 

.89 

.90 

.92 

.94 

.96 

68 

23 

.62 

.63 

.65 

.66 

.68 

.70 

.72 

.74 

.76 

.78 

.79 

.81 

.83 

.83 

.85 

.86 

.88 

.89 

.91 

.92 

.94 

.96 

.98 

1.00 

67 

24 

.65 

.66 

.68 

.69 

.71 

.73 

.75 

.77 

.79 

.81 

.83 

.84 

.85 

.87 

.88 

.90 

.91 

.93 

.94 

.96 

.98 

1.00 

1.03 

1.04 

66 

25 

.67 

.69 

.70 

.72 

.74 

.76 

.78 

.80 

.83 

.85 

.86 

.87 

.89 

.90 

.92 

.93 

.95  .96 

.08 

1.00 

1.02 

1.04 

1.06 

1.08 

65 

2G 

.70 

.71 

.73 

.75 

.76 

.78 

.80 

.83 

.85 

.88 

.89 

.90 

.92 

.93 

.95 

.97 

.98'l.00 

1.03 

1.04 

1.06 

1.08 

1.10 

1.12 

64 

27 

.72 

.74 

.75 

.77 

.79 

.81 

.83 

.86 

.88 

.91 

.92 

.94 

.95  .97 

.98 

1.00 

1.021.04 

1.05 

1.07 

1.09 

1.13 

1.14 

1.16 

03 

28 

.75 

.76 

.78 

.80 

.82 

.84 

.86 

.89 

.91 

.94 

.95 

.97 

.  98  1.  00 

1.02 

1.03 

1.05,1.07 

1.09 

1.11 

1.13 

1.15 

1.18 

1.20 

62 

29 

.77 

.79 

.81 

.82 

.84 

.87 

.89 

.91 

.94 

.97 

.98 

1.00 

1.  02 1.  03 

1.05 

1.07 

l.OO'l.  11 

1.13 

1.15 

1.17 

1.19 

1.22 

1.24 

61 

30 

.79 

.81 

.83 

.85 

.87 

.89 

.92 

.94 

.97 

1.00 

1.01 

1.03 

1.05  1.07 

1.08 

1.10 

1.121.141.16 

1.18 

1.21 

1.23 

1.25 

1.28 

60 

31 

.82 

.84 

.86 

.88 

.90 

.93 

.95 

.97 

1.00 

1.03 

1.05 

1.06 

1.  08,1. 10 

1.11 

1.13 

1.151.17 

1.20 

1.22 

1.24 

1.27 

1.29 

1.32 

59 

32 

.84 

.80 

.88 

.90 

.92 

.95 

.97 

1.00 

1.03 

1.06 

1.08 

1.09 

1.111.13 

1.15 

1.17 

1.19)1.21 

1.23 

1.25 

1.28 

1.30 

1.33 

1.36 

58 

33 

.87 

.88 

.91 

.93 

.95 

.97 

1.00 

1.03 

1.00 

1.09 

1.  U 

1.12 

1. 14  1. 16 

1.18 

1.20 

1.  22  1.  21 

1.26 

1.29 

1.31 

1.34 

1.37 

1.39 

57 

34 

.89 

.91 

.93 

.95 

.97 

1.00 

1.03 

1.05 

1.09 

1.13 

1.14 

1.15 

1. 17  1. 19 

1.21 

1.23 

1.  251.  27 

1.30 

1.32 

1.35 

1.37 

1.40 

1.43 

56 

35 

.91 

.93 

.95 

.98 

1.00 

1.03 

1.05 

1.08 

1.11 

1.15 

1.16 

1.18 

1.201.22 

1.24 

1.30 

1.291.31 

1.33 

1.36 

1.38 

1.41 

1.44 

1.47 

55 

36 

.93 

.95 

.98 

1.00 

1.03 

1.05 

1.08 

1.11 

1.14 

1.18 

1.19,1.21 

1.231.25 

1.27 

1.30 

1.321.34 

1.37 

1.39 

1.42 

1.45 

1.47 

1.51 

54 

37 

.96 

.98 

1.00 

1.02 

1.05 

1.08 

1.10 

1.14 

1.17 

1.20 

1.221.24 

1.261.28 

1.30 

1.33 

1.351.37 

1.40 

1.42 

1.45 

1.48 

1.51 

1.54 

53 

38 

.98 

1.00 

1.02 

1.05 

1.07 

1.10 

1.13 

1.16 

1.20 

1.33 

1.251.27 

1.291.31 

1.33 

1.36 

1.38'l.40 

1.43 

1.46 

1.48 

1.51 

1.54 

1.58 

52 

39 

1.00 

1.02 

1.05 

1.07 

1.10 

1.12 

1.15 

1.19 

1.22 

1.26 

1.281.30 

1.321.34 

1.36 

1.39 

1.411.43 

1.46 

1.49 

1.52 

1.55 

1.58 

1.61 

51 

10 

1.02 

1.04 

1.07 

1.09 

1.12 

1.15 

1.18 

1.21 

1.25 

1.29 

1.311.33 

1.351.37 

1.39 

1.42 

1.44  1.47 

1.49 

1.52 

1.55 

1.58 

1.61 

1.65 

50 

41 

1.04 

1.07 

1.09 

1.12 

1.14 

1.17 

1.20 

1.24 

1.27 

1.3l!l.331.35 

1.37 

1.40 

1.42 

1.45 

1.47 

1.50 

1.53 

1;55 

1.58 

1.61 

1.64 

1.68 

49 

42 

1.06 

1.09 

1.11 

1.14 

1.17 

1.20 

1.23 

1.26 

1.30 

1.  34jl.  361.  38 

1.40 

1.42 

1.45 

1.47 

1.50 

1.53 

1.55 

1.58 

1.61 

1.64 

1.68 

1.71 

48 

43 

1.08 

1.11 

1.13 

1.16 

1.19 

1.22 

1.25 

1.29 

1.32 

1.361.391.41 

1.43 

1.45 

1.48 

1.50 

1.53 

1.56 

1.58 

1.61 

1.64 

1.68 

1.71 

1.75 

47 

44 

1.10 

1.13 

1.15 

1.18 

1.21 

1.24 

1.28 

1.31 

1.35 

1.391.41,1.43 

1.46 

1.48 

1..50 

1.53 

1.56 

1.581.61 

1.64 

1.67 

1.71 

1.74 

1.78 

46 

45 

1.12 

1^ 

1.17 

1.20 

1.23 

1.26 

1.30 

1.33 

1.37 

1.411.441.46 

1.48 

1.51 

1.53 

1.56 

1.58 

1.611.64 

1.67 

1.70 

1.74 

1.77 

1.81 

45 

220 


A  MAXUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXVIII. — Factors  fur  reduction-  of  transit  ohserimtions — Continued. 

Aziiniith  factor  A  =  sill    sec,  5.    Star's  declination  ±  5.    Inclination  factor  B  =  cos  ^  sec.  5. 


i2°  53°  c 


i       51°  52° !  53°  54°  55°  50=  57°  58°  59°  '  60°  60^°  61°  61^°  02°  63|o  63°  63Jo  04°  64J°  05° 


46°  1. 14|1. 17  1. 19  1.  22  1.251.29 

47  |1.  lull.  19  1.211. 24  1.27J1.31 

48  1 1.  \&'y.  21  1.  23  1.  26  1 1.  30  1.  33 

49  1 1. 20|  1.  23  1.  25  1.  28  1.  32  1.  35 

50  1.221.24  1.  27  I.30I1.34I1.  3' 


1. 36  1.  40  1.  44il.  461 1. 48  1.  51|  1. 53  1.  56  1  58|1.  611.  64  1.  O'i  1.  70  1.  74  1.  77 
1.  38  1. 42!!  I.  40 1. 4911.  51 1. 53  1.  56  1.  58  1.  61,1.  61 1.  67  1'.  7ol  1.  73  1.  70  1.  80 
1. 40  1. 44,il.  48  1.  50,1.  53  1,55  1.  58  1.  60  1.  03  1.  66  1.  69  1.  72  1.  75  1.  79  1.  82 
1.  42  1,  47)1.  51 1.  53|1.  56  1.  58  11.  01 1.  63  1,  66  1.  60  1.  72  I.  75  .\.  79,1.  82  1.  86 
1.  44  1.  49;  1.  53  1.  5611.  58  1.  60  1.  03  1.  66  I.  69  1.  721.  75  1.  78|!1.  8111.  85  1.  88 


51  1.  J;  l.LM  l.-".i  l::-  1.:;.""'  l.:i  ij.4';  1-47  l.r.l  I. .".".  1.  .IS  1.  00  1.  63  1.  66  1.  08  1. 

52  l.•-'^  1.  J-  I  ::i  1  ;;  I,  ::  I.  (1  1.  -t:.  I  4:i  1.  :>:;  i.r,s  i.C'i  1. 11:;  1.65  1.68  1.71 1. 
5;i  1.27  1.:;  '  1  .  :  1.  :Vi  I.:::M.  I.:  I.  47  1..M  1.  '.'  l.i'.nl.  t:2  1.  ns  1.  07  |l.  70  1.  73  1. 

54  1.2:l  l.:;U.:U  l.::~  1.  41  1  4.'>  1.  4'i  1.  .) :  1.  ,'.7  1.  1)2  1.  04  1.  67  1.  09  1.72  1.75  1. 

55  1.30:1.33  1.36  1.30  1.4311.461.50  1.55  1.59  1.011.661.69  1.72  1.741.77  1. 


1.71  1.77  l.,-<i)  1..S4  1.871.91 

1.77  l.,sn  l..s::  I. so  1.90  1.94 

1.70  l..-<2  l.s",  1.  so  1.93  1.96 

l.sl  l.S,-.  l.S.S  1.911.951.99 

1.84  1.87  1.90  1.94  1.98  2.01 


63  ,1, 

04  1 

05  II, 


35  1.381.41 

36  1.  39  1.  43 
381.41  1.44 

39  1.421.40 

41  1.441.47  I. 

42  1.4.'"' 1.40  1. 
4:j  I.  47  1.. 'ill  1. 
45  1.40  1.52  1. 

40  1.  40  1.  5:;"  1. 
47  1.51  1.54  1. 


1. 45  I.  48 
1.46;  1.50 

1.48  1.52 

1.49  1.53 


!  1.50  1.61  1.661.081.711.74 
581.63  1.681.70|l.  731.  76 
60  1.65  1.701.7211.751.78 
02  1.66  1.711.7411.771.80 
631.  6.1'  1.  73  1.  76'1.  79  1.  81  tl.  84  1.  88  1.  911.  94  1.  97  2.  01  2.  05'2.  09  2. 13 


1.  77  1.  8.1 1.  83  1.  80  1.  89  1. 93  1.  90  2.  00  2.  04 
1.  79  1.  82  1.  85  1.  88  1.  91  1.  95  ll.  98  2.  02  2.  06 
1.  81  1. 84  1.  87  1. 90  1.  93  1.  97  |2.  01  2.  05  2.  08 
1.  83  1.  86  1.  89:1.  92  1.  95  1. 99  l2.  03'2. 07  2. 11 


.6411. 
.65  1. 
.66  1. 


18  1.73  1.781.81  1.84  LS7   l.liii  l.:i:;  l.'.i'  -      '  J 
'0,1.  75  ,1.80  1.  83  1.  85  1.  Ss  1.  Ill  1.  'X,  i.  0,-:  2.  U2  2.  "  J 
■1,1.  76  1 1. 81  1.  84]l.  87  1.  90  1.  93,1.  96  2.  00  2.  OJ  2.  07 


2.  11  2.15  2.19 
2.  13  2. 17  2.  21 
2. 14  2. 19  2.  23 


Oej°    67° 


1.87 
1.90 
1.93 


2.02 
2.04 
2.07 
2.10 


2.15 
2.17 
2.19 
2.22 


2.27 

.  20  2.  25  1  2.  29 
.  22  2.  20  ■  2.  31 
.24  2.28  2.32 
.  25  2.  30  ,  2.  34 
.  27  2.  31  I  2,  36 

71  'l.  50  1.  54  1.  57  1.  61  1.  65  1.69  1.  74  1.  78  1.  84  1.  80  1.  92  1.  95  1.  9S|  2.  01  2.  05  2.  08  2. 12  2. 10  2,  20  i2.  24  2.  28  2.  32    2.  37 

72  1 1.51  1.54  1.581.62  1.00  1.70  1.75  1.80  1.85  1.90  1.93  1.961.991 

73  !l.  52  I.  55  1.  .59  1.  63  1.  07  1.  71  1.  70  1.  SO  1.  86  1.91  1.  94  1.  97  2.  00;  2.  04  2.  07  2. 112. 14  2.  IS  2.  22  ;2.  26  2.  31  2.  35    2.  40 

74  1.53  1.56  1.60  1.63  1.081.72,1.701.811.87  1.92  1.  95  1.  98  2.  Oil  2.05  2.08  2. 12|2. 15  2. 19  2.  23 

75  11.  53  1.  57  1. 60  1.  64  1.  63  1.  73:1. 77:1. 82  1.  88  1.  93  1.  96  1.  99  2.  02l  2.  06  2.  09  2. 13i2. 16,2.  20  2.  24||2. 29|2.  33i2.  37  i  2.  42 


66  !  1. 45,1.  48  1.  52  1.  55  1.  .59  1.  03  1.  68  1.  721.  77  1.  83'l.  85' 1.  88  1.  Oil  1.  951.  98  2.  01'2.  05'2.  08  2. 12  I 

67  1.  46  1.  50  1.  53  1.  ,57  1.60  1.651.69  1.74  1.79  1.  S4  1.87  1.90  1.9:!  1.  9i;  1.  99  2.  0:i  2.  0(i  2.  10  2.  14 
03  1.47  1.511  54  1.  .58  1.621.06  1.70  1.75  1.8;;  1.S5  1.SS1.91  1.'.14  l.|i7  2.nl  j,  iin'  os  j.  11  j,  ]3 
69  1.481.521.551.59  1.63  1.071.711.701.81  1.87  1.9111.03  1.0:;  1.  Oil  2.  il2  J.  "n  2.  n:i  j.  ]:; -.  17 
JO  .1.49  1.53  1.50  1.00  1.041.68  1.73  1.771.82   1.  .SS  1.91  1.94  1.07  2.  OU  2.  u:;  2.  u7  2.  11  -.  14  2.  is 


2.32 

«2.34 
2.36 
2.37 
2.39 
2.40 

2.42 
2.43 
2.45 
2.46 
2.47 


76  1.54  1.58  1.611.05  1.  69  1.  73  1.781.  83;1.  SS  1.04  1.97  2.00  2.03' 

77  ll.  55  1.58  1.62  1.60  1.  70  1.  74  1.  79' 1.  84:1.  89  1.051,0-  2111  2  D  | 

78  ;l..55  !.. 59  1.621.66  1.701.  75I1.SO  1.85  1.90  1.06  1.:      .      :_     -j,.j-jj     -_ 

79  1.. 56  1.591. 631.67  1.71  1.701.80 1.S5  1.91  1.96  1.::   _      .  J  :     _        . 
8D    1.561.60  1.641.67  1.721.761.81,1.86  1.91  1.972.11  ■  J. II    -"     2     'J    :.:j    iT-: 

81   1.57|l.60'l.  64  1.68  1.72  1.77  1.81 1.80  1,02  1.  98  2. 111  j.  1:4  2.  "7  2.1:12.142  ls2.: 

jl.eill.  64 1.68  1.73  1.77  1.82  1.87  1.92  1.98  2.  "1  l."4  2.  us  -.  11  -J.  151,'.  IS  2.; 

I.  58  I.61I1.  65  1.69  1.73  1.77  1.  82  1.87  1.93  1.  90  2.  02  2.  05  2.  lis  -.  v^-l.  Vrl.  Iil2.: 

1.58ll.62  1.65l.69  1.  73  1.  78  1.  83!!.  88  1.  93  1.90  2.02  2.05  2.08  2.12  2,15  2,19  2,; 

1. 58  1.  62  1,  65  1,  69  11,  74  1,  78  1,  83  1, 83  1,  93  1,  99  2,  02  2,  0512,  091  2, 12,2, 16.2. 19|2. 1 


1.59  1.621,601,70  1,74  1,73  )..s:;l. 

1,  .59  1.  62  1.  66  1.70  1.  74  1.  7;i  1  -  :  i 

1.59  1.62  1.06  1.70  1.74  1  ' 

1.591.62  1.66  1,70  1,  74  ;  ^ 

1.591.621,661,70  1,74  i  T:'  1  ■,  ; 


n.94 


,  30'2. 34'2.  39  '  2.  43 

.  :n  2.  3r,  2.  40  2 

..  ..      ;;,;-  41)   ■_. 

_  _  :7  -.41  2 

:,  34  2,  38  2,  43  i  2. 48 
,  34(2,  39  2,  43  :  2,  48 
,  3512,  39  2,  44 
,  35|2,  40  2,  45 
.  3612. 40  2.  45 

,36  2,41,2,45 


2.49 


.  II  2.  40  2.  50  2,  50 

I  I  2  40  2.51  2,50 

I  2.40  2.51  2,; 

112,  40  2,  51  2,  56 


FACTOES  FOE  EEDUCTION  OF  TEAifSlT  OBSEKVATIONS.       221 


Table  XXYJIl.—Faeiors  for  reduction  of  transit  ohservations— Continued. 
Azimuth  factor  A  =  siu  i  sec.  S,    Star's  doclination  ±  S.    Inclination  factor  B  =  cos  i  sec.  5. 


i 

67J° 
.05 

68° 
.05 

68^0 
.05 

69° 
,05 

69io 
.05 

70° 
.05 

7040 
,05 

70J° 
.05 

.05 

71° 
.05 

... 

.05 

71i° 
.05 

711° 
.05 

72° 
.06 

72J° 
.06 

72i° 
.06 

72JO 
.06 

73° 
.06 

73i° 
.06 

73i° 
.06 

73i° 
.06 

740 
.06 

74i° 
.06 

i 

89° 

1° 

2 

.09 

.09 

.10 

,10 

.10 

.10 

,10 

.10 

.11 

.11 

.11 

.11 

.11 

.11 

.11 

.12 

.12 

.12 

.12 

.12 

.12 

.13 

.13 

88 

3 

.  14 

.14 

.  14|  .  15    .  15 

.15 

,15 

.16 

.16 

.16 

.16 

.16 

.17 

.17 

.17 

.17 

.18 

.18 

.18 

.18 

.19 

.  19 

.19 

87 

l 

.18 

.19 

.19    .20    .20 

.20 

.21 

.21 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.23 

.23 

.24 

.24 

.24 

.25 

.25 

.26 

86 

5 

.23 

.23 

.24    .24 

.25 

.25 

.26 

.26 

.26 

.27 

.27 

.27 

.28 

.28 

.29 

.29 

.29 

.30 

.30 

.31 

.31 

.32 

.32 

85 

6 

,27 

.28 

.28   .29 

.30 

.31 

.31 

.31 

.32 

.32 

.33 

.33 

.33 

.34 

.34 

.35 

.35 

.36 

.36 

.37 

.37 

.38 

.39 

84 

7 

.32 

.33 

.33!  .34 

.35 

.36 

.36 

.37 

.37 

.37 

.38 

.38 

.39 

.39 

.40 

.41 

.41 

.42 

.42 

.43 

.44 

.44 

.45 

83 

8 

.36 

.37 

.38  .39 

.40 

.41 

.41 

.42 

.42 

.43 

.43 

.44 

.44 

.45 

.46 

.46 

.47 

.48 

.48 

.49 

.50 

.50 

.51 

82 

g 

.41 

.42 

.43    .44 

.45 

.46 

.46    .47 

.47 

.48 

.49 

.49 

.50 

.51 

.51 

.52 

.53 

.53 

.54 

.55 

.56 

.57 

.58 

81 

10 

.45 

.46 

.47|  .49 

.50 

.51 

.61    .52 

.53 

.53 

.54 

.55 

.55 

.56 

.57 

.58 

.59 

.60 

.60 

.61 

.02 

.63 

.64 

80 

11 

.50 

.51 

.52'  ..53 

.54 

.56 

.56    .57 

.58 

.59 

.59 

.60 

.61 

.62 

.63 

.63 

.64 

.65 

.66 

.67 

.68 

.69 

.70 

79 

12 

.54 

.56 

.57 

,58 

.59 

.61 

.62    .62 

.63 

.64 

.65 

.63 

.66 

.67 

.68 

.69 

.70 

■.71 

.  72 

.73 

.74 

.75 

.77 

78 

13 

.59 

.60 

.61 

,63 

.64 

.66 

.67 

.67 

.68 

.69 

.70 

.71 

.72 

.73 

.74 

.75 

.76 

.77 

.78 

.79 

.80 

.82 

.83 

77 

14 

.63 

.65 

.66 

.68 

.69 

.71 

.72 

.72 

.73 

.74 

.75 

.76 

.77 

.78 

.79 

.80 

.82 

.83 

.84 

.85 

.87 

.88 

.89 

76 

15 

.68 

.69 

.71 

.72 

.74 

.76 

.77 

.78 

.78 

.79 

.80 

.81 

.83 

.84 

.85 

.86 

.87 

.89 

.90 

.91 

.93 

.94 

.95 

75 

16 

.72 

.74 

.75 

.77 

.79 

.81 

.82 

.83 

.84 

.85 

.86 

.87 

.88 

.89 

.91 

.92 

.93 

.94 

.96 

.97 

.99;  1.00 

1.02 

74 

17 

.76    - 7S 

.  80'  .  HI 

.83 

.85 

.86'  .88 

.89 

.90 

.91 

.92    .93 

.95 

.96 

.97 

.99 

1.00    1.01 

1.03 

1.05!  1.06 

1.08 

73 

18 

.81 

.83 

.84|  .86!  .88 

.90 

.911  .93 

.94 

.95 

.96 

.97    .991.00  1.01 

}-??o 

1.04 

1.06 

1.07 

1.09 

1. 10    1.  12 

1.14 

72 

19 

.85 

.87 

.89'  .91    .93 

.95 

.96i  .98 

.99 

i.oo!i.oi 

1.  03  1.  04  1.  05 

1.07 

1.08 

1.10 

1.11 

1.13 

1.15 

1. 16i  1.18 

1.20 

71 

20 

.89 

.91 

.93    .95    .98 

1.00 

1.011.02 

1.04 

1.05:1.06 

1.081.091.11 

1.12 

1.14 

1.15 

1.17 

1.19 

1.20 

1.22j  1.24 

1.26 

70 

21 

.94 

gii 

.  98  1.  00  1.  02 

1.  05 

1.06:1.07 

1.09 

1,10  1,11  l,13  1,14'l,16 

1.17 

1.19 

1.21 

1.22 

1.24 

1.26 

1.28   1.30 

1.32 

69 

22 

.981.00 

1.021.051.07 

l.OO.l.ll'l.  12 

1.14 

1,15  1,17  1,  LSI,  20  1.2111,  23 

1.25 

1.26 

1.28 

1.30 

1.32 

1.34;  1.36 

1.38 

68 

23 

1.021.04 

1.07  1.091.12 

1,  14!l,161,17 

1.19 

1,2(11   21    1,2:;  1,25  1,26  1,28 

1.30 

1.32 

1.34 

1.36 

1.38 

1.40'  1.42 

1.44 

67 

24 

1.  Oe'l.  09 

1.  11  1.  14  1.  Ifi 

1. 19  1.  20  1,  22 

1,  23  1,  li.'.  1.  27  1 ,  i.l  1,  30  1,  32  1,  33|1,  35 

1.37 

1.39    1.41 

1.43 

1.45   1.48 

1.50 

66 

25 

l!l0|l.l3 

1.1,-.  1,1s  1,21 

1,241,251,27 

1.28 

l.i'.ii  1,31  1,:;:!  1.  3,' 1,37  1,39,1,41 

1.42 

1.45,  1.47 

1.49 

1.51 

1.53 

1.56 

65 

26 
27 
28 
29 
30 

1. 15  1. 17 

,1,1        ,1,1        .ir, 

1,  28  1,  30' 1,31 

1,  Sj 

1.:;,-.  l.:;ii  1.  :i,-<  1.411 1,421,  ll'l,  46 

1.48 

1.51 

1.52 

1.54 

1.57 

1.59 

1.61 

64 

l!l9l'-Jl 

1.  •_*-!  1.  'JT  1,  :!0 

1.  :!;i'l,  34  1,  31 

1.  :;.^ 

l.:;:i  1,  11   1,  1:;  1. 4.' 1. 47  1, 49 1.  r.l 

1.53 

1.  55 

1.58 

1.60 

1.62 

1.65 

1.67 

63 

1.  ■j,'^  1,  ;ii  1 .  ,14 

1,  :;7  1,  :;'.i !.  11 

1,  IL 

1,  .|.|  1.  Ill    1.4^  1.  .'Hi  1.  .■•2  1,  Til  1.  -'1' 

1.58 

1.00 

1.63 

1.65 

1.68 

1.70 

1.73 

62 

1     '"^~  1     ■"'! 

1.  41;  1,  l:M  .  r 

1. 47 

1  .j'l  1  .■.  1    I  :■:]  )  .'.',  1.  ,",7  1,  .''11  1  111 

1.63 

1.661  1.68 

1.71 

1.73 

1.76 

1.79 

61 

L3u'..5:; 

l,:j(il,:;9  1,4:1 

1,40  L4,S  L.-'l 

1.  nil.. -.11  l..-,s  1,111)  1.112  1,114  1,110 

1.69 

1.71    1.73 

1.76 

1.79 

1.81 

1.84 

60 

31 

1.35 

1.38 

1.40'l.44'l,47 

'1.51  1,52  1,54 

1,56 

1,  58  1.  60  1,  62  1,  64  1,  67  1,  69|1, 71 

1.74 

f.iel  1.79 

1.81 

1.84 

1.87 

1.90 

59 

32 

1.39 

1.421.451.481.51 

1.  55!l.  57  -I.  59 

1,61 

1,03  1.65' 11.67  1,691,711,74 

1.7b 

1.79 

1.81    1.84 

1.87 

1.89 

1.92 

1.95 

58 

33 

1.42 

1.451.49  1.521.55 

'1.59;l,0i:i,63 

1,65 

1,67,1,69 

!l.721.741.761.7£ 

1.81 

1.84 

1.86    1.89 

1.92 

1.95;  1.98 

2.01 

57 

34 

1.46 

1 .  49  1 .  53  1 .  56  1 .  60 

1,  63]l,  65  1.  6S 

1,70 

1.  72il.  74 

'l.7G1.791.811.8r 

l.Sb 

1.80 

1.91]  1.94 

1.97 

2.00    2.03 

2.06 

56 

35 

1.  50|1.  53 

1,56,1.601,64 

|1,  681,701,72 

1,74 

,1.761.78 

1.81|1.831.  861.8f 

1.91 

1.93 

1.961  1.99 

2.02 

2.05    2.08 

2.11 

55 

36 

1.54 

1.57 

1,  60  1.  64  1,  68 

1         1 
1.721.74  1.76 

1,78 

1.80  1.83 

11.851.881.901.9 

1.95 

1.98 

2.01    2.04 

2.07 

2.10 

2.13 

2.16 

54 

37 

1.57 

1.  61 

1,  64  1,  68  1,  72 

1,761,  78  1,80 

1,83 

'1.851.87 

1  90  1.  92  1.  95  1.  97i2.  00 

2.03 

2.06    2.09 

2.12 

2.15 

2.18 

2.22 

53 

38 

1.61 

1.  64 

1.  681.721,76 

1,  80  1,  82  1.  84 

1,87 

1.891.91 

1.941.971.99  2.02  2.05 

2.08 

!  2.11    2.14 

2.17 

2.20 

l^i 

2.27 

52 

39 

1  65 

1  68 

1.721,751,80 

1,84  1,861,88 

1,91 

1.931.96 

1.98  2.012.04:2.06  2.09 

2.12 

,  2.15    2.  IE 

2.22 

2.25 

2.28 

2.32 

51 
50 

40 

1.68 

1.72 

1,751,791,84 

1,881,901,93 

1.95 

1.97  2.00 

2.  03  2.  05  2.  08  2. 11  2. 14 

2.17 

2.20    2.23 

2.26 

2.3C 

2.83 

2.37 

41 

1.71 

1.75 

1.79  1.83  1,87 

l,92'l,94  1.9f 

i.or 

2.  01  2.  04 

2.  07  2.  09!2. 12  2. 15!2.18 

2.21 

2.24  2.2E 

2.31 

2.  84 

2.38 

2.42 

49 

42 

1.75 

1.79 

1,  S3  1,  87  1,  91 

1,96,1,  9H2,  01 

2.  0: 

2.  05  2. 118,  2. 11  2. 14'2.  16  2.  19!2.  22 

2.26 

2.2 

2.31 

2.3e 

2.3£ 

2.43 

2.46 

48 

43 

1.  7f 

1.82 

1,  86  1,  90  1,  95 

1,99  2,02  2,04 

2.  0" 

2.  09  2.  1-. 

2.  15  2.182.  212.  24!2.  27 

2.30 

2.3 

2.3'" 

2.4t 

2.41 

2.47 

2.51 

47 

44 

1  82 

1  85 

1,  90  1.  94  1  98 

2,  03:2,  06  2.  08 

2,1 

i2. 13  2. 16 

2. 19  2.  22  2.  25  2.  28!2.  31 

2.34 

2.3 

2.41 

2.4E 

2.4f 

2.52 

2.5e 

46 

45 

1.85 

1.89 

1.931,97  2,02 

2,  07  2.  09  2.  12,2, 1 

1         1 

;2.17i2.20 

1 

2.23  2.26  2.29,2.32  2.36 

1        !        1 

2.  88 

2.4 

2.4> 

2.4E 

2.5 

2.5* 

2.  6C 

45 

222 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXVIII. — Factors  for  reduction  of  transit  oiservations — Continued. 
Azimuth  factor  A  =  sin  J  sec.  S,    Star's  declination  ±  8.    Inclination  factor  B=co3  i  sec.  i. 


i 

67S° 

l.SS 

68° 
1.92 

68io 
1.96 

69° 
2.01 

69io 
2.05 

70° 
2.10 

70J° 
2.13 

704° 
2.15 

703° 
2.18 

71°  71J' 

71J° 

71P 
2.30 

72° 
2.33 

72J° 
2.36 

72i° 
2.39 

72J° 
2.42 

73° 
2.46 

731° 
2.49 

73JO 
2.53 

73i° 
2.57 

74° 
2.61 

7^140 
2.65 

? 

46° 

2.212.24 

44° 

47 

i.iiiii.ti:. 

2.  no  -J.  (14 

2.  Oil 

2.14 

'  111  ■'  I'.i  "  ■ :.  ■'  ",  ■'  :;n 

3.33 

2.37 

2.40 

2.43 

2.47 

2.50 

2.54 

2.57 

2.61 

2.65 

2.69 

43 

48 

l-'.l-l  L'.  us 

2.  I'2  2.  07 

2.  12 

2.  17 

2,111  J    -J.'    L-    .    -■-■     :i   -     :12.37 

2.40 

2.44 

2.47 

2.51 

2.54 

2.58 

2.62 

2.66 

2.70 

2.74 

42 

49 

l.L'T  -.  "1 

2.  (Hi  2.  ]  1 

2.  11; 

2.21 

'■'::■■                            :.       ;s  ■"*  41 

2.44 

2.48 

2.51 

2.55i 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

41 

50 

1.  (JO  2.  Ul 

2.  US  2. 14 

2.1a 

2.24 

2.  27,2.  2.  2.  .2,  2.  .3,2...,     11,2.45 

2.48 

2.51 

2.55 

2.58 

2.62 

2.66 

2.70 

2.74 

2.78 

2.82 

40 

51 

2.03 

2.07 

2. 12  2. 17 

2.22 

2.27 

2.30  2.33 

2.36 

2.39 

2.42 

2.45 

2.48 

2.51 

2.55 

2.58 

2.62' 

2.66 

2.70 

2.74 

2.78 

2.82 

2.86 

39 

52 

2.06 

2.10 

2.15 

2.20 

2.25 

2.30 

2.33  2.36 

2.39 

2.42 

2.45 

2.48 

2.52 

2.55 

2.58 

2.62 

2.66, 

2.69 

2.73 

2.77 

2.82 

2.86 

2.90 

38 

53 

2.09 

2.13 

2.18 

2.23 

2.28 

2.33 

2. 36  2. 39 

2.42 

2.45 

2.48 

2.52 

2.55 

2.58 

2.  62  2. 66 

2.69 

2.73 

2.77 

2.81 

2.85 

2.90 

2.94 

37 

54 

2.11 

2.16 

2.21 

2.26 

2.31 

2.37 

2.39  2.42 

2.45 

2.48 

2.52 

2.55 

2.58 

2.62 

2.  65  2.  69 

2.73 

2.77 

2.81 

2.85 

2.89 

2.94 

2.98 

36 

55 

2.14 

2.19 

2.23 

2.29 

2.34 

2.40 

2.42  2.45 

2.48 

2.52 

2.55 

2.58 

2.62 

2.65 

2.69  2.72 

2.76 

2.  SO 

2.84 

2.88 

2.93 

2.97 

3.02 

35 

56 

:.i7-:.2i 

2  ^7 

2  42 

2.15  2  1  = 

2  51   2  55  2  5«'2  01 

2  fi5'2,  m 

2.  72  2.  76 

2.80 
2.83 
2.86 

2.84 
2.87 
2.90 

2.88 
2.91 
2.94 

2.92 
2.95 
2.99 

2.96 
3.00 
3.03 

3.01 
3.04 
3.08 

3.05 
3.09 
3.12 

34 
33 
32 

57 

-  '" '  -' 

'-  ■,  _  :.,  -  ■:  'J :    -  '  :  ;  i  -^  :i  I  742:78|2!82 

58 

59 
50 

I... .11  42 

2.47 

2w.j 

^  7,  J  77  _  .  ..  _  .  ..  ..-r.  „   7.._,  ;  ,  ..77  2.8ll2.85 

2.89 
2.92^ 

2.93 
2,96 

2.97 
3.01 

3.02 
3.05 

3.06 
3.09 

3.11 
3.14 

3.16 
3.19 

31 
30 

liioli.Jl 

-..Jl,-.obl2.0.,   2.(jU2.(,i)2.  ,o2.  ,0 

2.80 

2.84  2.88 

61 

2. 2912.  .33 

2.  39  12. 44 

2.50 

2.56 

2.59  2.62  2.65  2.69  2.72  2.76 

2  79 

2.83 

2.8712.91 

2.95 

2.99 

3.04 

3.08 

3.13 

3.17 

3.22 

29 

62 

2.312.36 

2.41:2.46 

2.52 

2.58 

2.612.64  2.68  12.712.75  2.78 

2  82 

2.86 

2.90  2.94 

2.98 

3.02 

3.06 

3.11 

3.16 

3.20 

3.25 

28 

63 

2.33;2.38 

2.43  2.49 

2.54 

2.6(1 

2.  64  2.  67  2.  70'  2.  74  2.  77  2. 81 

2.84 

2.88 

2.92I2.96 

3.00 

3.05 

3.09 

3.14 

3.18 

3.23 

3.28 

27 

64 

2.35|2.40 

2.45  2.  SI 

2.  rt~ 

2.  C3 

2.  66  2.  69  2.  73  2.  76  2.  80i2.  83 

2.87 

2.91 

2.9512.99 

3.03 

3.07 

3.12 

3.16 

3.21 

3.26 

3.31 

26 

65 

2.37  2.42 

2. 47  2.  52 

2.59 

2.65 

2.88  2.71,2.75,2.78,2.82 

2.86 

2.89 

2.93 

2.  97j3.  01 

3.06 

3.10 

3.14 

3.19 

3.24 

3.29 

3.34 

25 

66 

2,  I'l 

2  1-- 

2,7"2.  74  2,  77  2,1-12.84 

2.88 

2.92 

2.96 

3.00  3.04 

3.08 

3.13 

3.17 

3.22 

3.27 

3.31 

3.37 

24 

67 

-!_-■.   -:■   .    -12.86 

2.90 

2.94 

2.98  3.02,3.06 

3.10 

3.15 

3.20 

3.24 

3.29 

3.34 

3.39 

23 

68 

7        7-       -:    _    --iL7  8Si2.92 

2.96 

3.0ol3.04i3.08 

3.13 

3.17 

3.22 

3.26 

3.31 

3.36 

3.42 

22 

69 

J .  -U  '^ .  4 

J  7',  J  -ii  J   -  1  ll,,-7  2.90]2.94 

2.98 

3.0213.06  3.10 

3.15 

3.19 

3.24 

3.29 

3.34 

3.39 

3.44 

21 

70 

.'.  4U  ■.;.  51 

2.  oli  2.  Ui; 

2.  US 

2.  iS  2.  81  2.  8.5  2.  89,2.  92,2.  96 

3.00 

3.04(3.08  3.12 

3.17 

3.21 

3.26 

3.31 

3.36 

3.41 

3.46 

20 

71 

2.47 

2.52 

2.58 

2.64 

2.70 

2.77 

2.8o'2.83'2.87'2.90  2,94  2.98 

3.02 

3. 06^3. 103. 14 

3.19 

3.24 

3.28 

3.33 

3.38 

3.43 

3.48 

19 

72 

2.49 

2.54 

2.59 

2.65 

2.72 

2.78 

2. 8Ij2.  85j2.  88  :2.  92:2.  96'3.  00i3.  04 

3.  08,3. 12,3. 16 

3.21 

3.25 

3.30 

3.35 

3.40 

3.45 

3.50 

18 

73 

2.50 

2.55 

2.61 

2.67 

2.73 

2.80 

2. 83(2.  8612.  90l  2.  94  2.  97  3.  0l|3.  05 

3.09  3.14  3.18 

3.22 

3.27 

3.32 

3.37 

3.42 

3.47 

3.52 

17 

74 

2.51 

2.57 

2.62 

2.68 

2.74 

2.81 

2. 84,2. 88  2.  92;  2.  95  2.  99  3.  03,3.  07 

3.ir3.]5  3.20 

3.24 

3.29 

3.33 

.3.38 

3.44 

3.49 

3.54 

16 

75 

2.52 

2.58 

2.64 

,2.70 

2.76 

2.82 

2. 86|2. 89l2. 93|  2.  97,3.  00,3.  04 

3.08 

3.13  3.17  3.21 

3.26 

3.30 

3.35 

3.40 

3.45 

3.50 

3.56 

15 

76 

J..MJ. -;- 

2.  i:-'2.  71 

2.  -1 

2  -7  11  111  -M7  J  nil  11. 02:3. 06 

3.10 

3.15,3.18:3.23 

3.28 

3.32 

3.37 

3.42 

3.47 

3.53 

3.58 

14 

-    -■        -  -      "    -   I'l'::.  0313.07 

3.11 

3. 1513.19  8.  24 

3.29 

3.33 

3.38 

3.43 

3.48 

3.54 

3.59 

13 

78 

.     J        .   ,  -         "11. 04'3.08 

3.12 

3.16,3.213.25 

3.30 

3.34 

3.39 

3.44 

3.49 

3.55 

3.60 

12 

79 

-■'  i;^  ■'  7-1 

J,  111  -.iMu,'K  :i..i-:i  (1513.09 

3. 13|3. 18l3.  22 

3.26 

3.31 

3.36 

3.41 

3.46 

3.51 

3.56 

3.62 

11 

80 

2.  57  2.  63 

2!  69  2. ::, 

2.  81 

2^88 

2.  91  2.  95  2.  99  3.  02  3.  06  3. 10 

3.14 

3. 19  3.  23 

3.27 

3.32 

3.37 

3.42 

3.47 

3.52 

3.57 

3.63 

10 

81 

2. 58' 2.  64 

2. 69  2.  76 

2.82 

2.89 

2.  92  2.  9613.0013.  03  3.07:3.11 

3.15 

3.  20  3.  24 

3.28 

3.33 

3.38 

3.43 

3.48 

3.53 

3.58 

3.64 

9 

82 

2.59,2.64 

2.70  2.76 

2.83 

2.90 

2. 93  2.  97  3.  00  3.  04  3.  08  3. 1213. 16 

3.  20  3.  25i3.  29 

3.34 

3.39 

3.44 

2.49 

3.54 

3.59 

3.65 

8 

83 

2.59  2.65 

2.71  2.77 

2.83 

2.90 

2.  94  2.  97  :;.  Ill  2.  0,5  3.  0913. 13'3. 17 

3.21,3.2613.30 

3.35 

3.40 

3.45 

3.49 

3.  55 

3.60 

3.66 

7 

84 

2.  60  2.  66 

2.71:2.78 

2.84 

2.91 

2.94  2.11,-11.112  li,  ml  n.  09'3. 13  3. 18 

3.22  3.263.31 

3.35 

3.40 

3.45 

3.50 

3.55 

3.61 

3.66 

6 

85 

2.  60  2.  66 

2.72i2.78 

2.84 

2.91 

2.  95  2,  98  IJ.  U2  11.  Ui;  3. 1013. 14:3. 18 

3.  22  3.  27(3.  31 

3.36 

3.41 

3.46 

3.51 

3.56 

3.61 

3.67 

5 

86 

2.  61  2.  l-J 

2.92 

2.95 

2.99,3.03 

.3.  06|3. 10 

3. 14  3. 19 

3.  23  3.  27 

3.32 

3.36 

3.41 

3.46 

3.51 

3.57 

3.62 

3.68 

4 

87 

2.612.1' 

2.92 

2.95 

2.99 

3.03 

3.07 

3.11 

3. 15  3. 19 

3.  23  3.  28 

3.32 

3.37 

3.42 

3.47 

3.52 

3.57 

3.62 

3.68 

3 

88 

2.61  2.  i;: 

2.92 

2.96 

2.-99 

3.03 

3.07 

3.11 

3.15  3.19 

3.23  3.28 

3.32 

3.37 

3.42 

3.47 

3.52 

3.57 

3.62 

3.68 

2 

89 

2. 6112.  u: 

2.  71>||2.  71) 

2.80 

2.92 

2.96 

3.00 

3.03 

3.07 

3.11 

3.15  3.19 

3. 24  3.  28 

3.33 

3.37 

3.42 

3.47 

3.52 

3.57 

3.63 

3.68 

1 

90 

2.612.67 

2.  73|J2.  79 

:f 

2.92 

2.96 

3.00 

3.03 

3.07 

3.11 

3. 15  3. 19 

3.24  3.28 

3.33 

3.37 

3.42 

3.47 

3.52 

3.57 

3.63 

3.68 

0 

FACTOES  FOE  EEDUCTION  OF  TEANSIT  OBSERVATIOifS.       223 


Table  XXVIII. — Factors  for  reduction  of  transit  observations — Continued. 
Azimuthfactor  A=:8inisec.  6.     Star's  tleclination  ±  S.    Inclination  factor  B  =  cos  ^  sec.  S. 


i 
1^ 

74i° 
.06 

74,o 
.07 

75° 
.07 

751^ 
.07 

7=S° 
.07 

75,0 

76° 

76,° 
.07 

76,° 
.07 

76»° 
.08 

77° 
.08 

77,° 
.08 

77i° 
.08 

77i° 
.08 

78° 
.08 

78i° 
.09 

78J° 

78J° 

,  79° 

79i° 
.09 

79J° 
.10 

79J° 
.10 

80° 
.IC 

i 

.07 

.07 

.09 

.  09:     •  09 

'89 

2 

.13 

.13 

.13 

.14 

.14 

.14 

.14 

.15 

.15 

.15 

.16 

.16 

.16 

.16 

.17 

.17 

■  .18 

.  18'     .  16 

.19 

.19 

.20 

.2C 

88 

3 

.20 

.20 

.20 

.21 

.21 

.21 

.22 

.22 

.22 

.23 

.23 

.24 

.24 

.25 

.25 

.26 

.26 

.  271     .  27 

.28 

.29 

.29 

.31 

87 

4 

.26 

.27 

.27 

.27 

.28 

.28 

.29 

.29 

.30 

.30 

.31 

.32 

.32 

.33 

.34 

.34 

.35 

.  36:     .  37 

.37 

.38 

.39 

.41 

86 

5 

.33 

.33 

.34 

.34 

.35 

.35 

.36 

.37 

.37 

.38 

.39 

.40 

.40 

.41 

.42 

.43 

.44 

.45 

.46 

.47 

.48 

.49 

.50 

85 

6 

.39 

.40 

.40 

.41 

.42 

.42 

.43 

.44 

.45 

.46 

.46 

.47 

.49 

.49 

.51 

.51 

.52 

.54 

.55 

.56 

.57 

.59 

.60 

84 

7 

.46 

.46 

.47 

.48 

.49 

.50 

.50 

.51 

.52 

.53 

.64 

.55 

.56 

.57 

.59 

.60 

.61 

.62 

.64 

.65 

.67 

.69 

.70 

83 

8 

.52 

.53 

.54 

.55 

.56 

.57 

.58 

.59 

.60 

.61 

.62 

.63 

.64 

.66 

.67 

.68 

.70 

.71 

.73 

.75 

.76 

.78 

.80 

82 

9 

.58 

.59 

.60 

.61 

.62 

.64 

.65 

.66 

.67 

.68 

.70 

.71 

.72 

.74 

.75 

.77 

.78 

.80 

.82 

.84 

.86 

.88 

.90 

81 

10 

.65 

.66 

.67 

.68 

.69 

.71 

.72 

.73 

.74 

.76 

.77 

.79 

.80 

.82 

.84 

.85 

.87 

.89 

.91 

.93 

.95 

.98 

1.00 

80 

11 

.71 

.73 

.74 

.75 

.76 

.77 

.79 

.80 

.82 

.83 

.85 

.86 

.88 

.91) 

.92 

.94 

.96 

.98 

1.00 

1.02 

1.05 

1.07 

1.10 

79 

12 

.78 

.70 

.80 

.82 

.83 

.85 

.86 

.88 

.89 

.91 

.92 

.94 

.96 

.98 

1.00 

1.02 

1.04 

1.07 

1.09 

1.11 

1.14 

1.17 

1.20 

78 

13 

.84 

.86 

.87 

.88 

.90 

.91 

,93 

.95 

.96 

.981.00  1.02 

1.04 

1.06 

1.08 

1.10 

1.13 

1.15 

1.18 

1.21 

1.23 

1.26 

1.30 

77 

11 

.91 

.92 

.94 

.95 

.97 

.98 

1.00 

1.02 

1.01 

1.061.08  1,10 

1.12 

1.14 

1.16 

1.19 

1.21 

1.24 

1.27 

1.30 

1.33 

1.36 

1.39 

76 

15 

.97 

.98 

1.  Oil 

1.02 

1.03 

1.05 

1.07 

1.09 

1.11 

1.13  1.15  1,17 

1,20 

1.22 

1.25 

1.27 

1.30 

1,33 

1.36 

1.39 

1.42 

1.46 

1,49 

75 

16 

1.03 

1.05 

1.06 

1.08 

1.10 

1.12 

1.14 

1, 16  1. 18'i,2o'l,2:i  1.25 

1.28 

1.30 

1.33 

1.35 

1.38 

1.41 

1.44 

1,48 

1.51 

1,55 

1,59 

74 

17 

1.  09 

1,11 

1.13 

1.15 

1.17 

1.19 

1.21 

1,  2:l!l,  25  1.  28  1,  30  1.  32 

1.35 

1,38 

1.40 

1.44 

1.47 

1.50 

1.53 

1.57 

1.60 

1,64 

1.68 

73 

19 

1.16 

1.17 

1.19 

1.21 

1.23 

1.  25  ;i.  2S  1.  3U|1,  32  1.  35  1.  37  1,  411 

1.43 

1.46:i.49 

1.52 

1.55 

1.58 

1.62 

1.66 

1.70 

1.74 

1.78 

72 

10 

1.22 

1.24 

1.20 

1.28 

1.30 

1.32 

1.  .35 

1.37  1,  39|1,  42:1,45  1.47 

1.51 

1.  53 

1.57 

1.60 

1.63 

1.67 

1.71 

1.75 

1.79 

1.83 

1.87 

71 

20 

1.28 

1.30 

1.32 

1.34 

1.37 

1.30 

1.41 

1.  44ll,461,  49  1,50  1.55 

1.58 

1.61 

1.65 

1.68 

1,73 

1.75 

1.79 

1.83 

1.88 

1.93 

1.97 

70 

21 

1.34 

1.36 

1.3R 

1.41 

1.43 

1.46 

1.48 

1,51 1,541,  56  1.591  1.62 

1.65 

1.69 

1,72 

1.76 

1.80 

1.84 

1.88 

1.92 

1.97 

2.01 

2.06 

69 

22 

1.40 

1.42 

1.45 

1.47 

1.50 

1.52 

1.  55 

1.  58  1,  60  1,  63  1.  66'  1.  ■Jo 

1,73 

1.77 

1,80 

1.84 

1.88 

1.92 

1.96 

2.01 

2.06 

2.11 

2.16 

68 

23 

1.46 

1.49 

1.51 

1.54 

1.56 

1.59 

1.62 

1.  04  1.  67  1.  70  1.  74[  1.  77 

1,81 

1,  84il.88 

1.92 

1.96 

2.00 

2.05 

2,09 

2.14 

2.20 

2.25 

67 

24 

1,  .52 

1.55 

1.57 

1.60 

1.63 

1.65 

1.68 

1,711,  74jl,  771.  8ll|l,  84 

1,88 

1.  92  1.  96 

2.00 

2.04 

2.08 

2.13 

2.18 

2.23 

2.29 

2.34 

66 

25 

1.58 

1.61 

1.63 

1.66 

1.69 

1.72 

1.75  1.781,811,84  1.88  1,91 

1.95 

1,99  2.03 

2.07 

2.12 

2.17 

2.22 

3.27 

3.32 

2.38 

2.43 

65 

26 

1.64 

1.67 

1.69 

1.72 

1.75 

1.78 

1.811.841,881,911,9.-1  1,9!) 

2.  02 

2,  07:2. 11 

2.15 

2.20 

2.25 

2.31 

2.35 

2.41 

2.46 

2,52 

64 

27 

1.70 

1.73 

1.75 

1.78 

1.81 

1.85 

1,  88ll.911.951.98  2,  (12  2,  111! 

2,  10 

2,14'2.18 

2.23 

2,28 

2.33 

2.38 

2,43 

2.49 

2.55 

2,61 

63 

28 

1.70 

1.78 

1.81 

1.84 

1.87 

1.91 

I.  94  1.  97  2.  0i;2.  05  2, 119  2. 13 

2,17 

2,  21 '2,  26 

2.31 

2.36 

2,41 

2.46 

2,52 

2'.  58 

2.64 

2.70 

62 

29 

1.81 

1.84 

1.87 

1.90 

1.94 

1.97 

2.  002.04  2.  08.2. 112. 15i!2.  20 

2.24 

2,  28  2.  33 

2.38 

2.43 

2,48 

2.54 

2.60 

2.66 

2.73 

2.79 

61 

30 

1.87 

1.90 

1.93 

1.96 

2.00 

2.03 

2.07:2.  10  2. 14:2. 18  2.  22!i2.  27 

3.31 

2.36  2.40 

2.46 

2.51 

2.56 

3.62 

2.68 

2.74 

2.81 

2.88 

60 

31 

1.93 

1.96 

1.99 

2.02 

2.06 

2.09 

2. 13  2.  17  2.  2rj  J.V-J. -J'.i  J.  :::; 

2,38 

2.  43  2  48 

2.53 

2.58 

3.64 

2.70 

2.76 

2.83 

2.89 

2.97 

59 

32 

1.98 

2.01 

2.05 

2.08 

2.12 

2.15 

2, 19  2, 23  2, -j:  :       -    '    :   ,1 

2,45 

2.50  2.55 

2.60 

2.66 

2.72 

2.78 

2.84 

2.91 

2.98 

3.05 

58 

33 

2.04 

2.07 

2.10 

2.14 

2.18 

2.21 

2.25  2.29  2,:::;-     -  -   i : 

2,52 

2.57i2.62 

2.67 

2.73 

2.79 

2.85 

2.93 

2.99 

3.06 

3.14 

57 

34 

3.09 

2.13 

2.16 

2.20 

2.23 

2.  27 

2.312,35  2,4111'    1  '  :    I  '  J.:.:: 

2,58 

2.  64'2.  69 

2.75 

2.80 

2.87 

2.93 

3.00 

3.07 

3,14 

3,23 

56 

35 

2.15 

2.18 

2.22 

2.25 

2.29 

2.33 

2,37  2.412,40  J.  riij,  l.'i  J.r.n 

2,65 

2.  70  2.  76 

2.82 

2.88 

2.94 

3.01 

3.08 

3,15    3.28 

3.30 

55 

36 

2.20 

2.24 

2.27 

2.31 

2,35 

2.39 

2.432.472,,52-J.,-i;2.i;l   2.  CO 

2,  77  2.  83 

2.89 

2.95 

3.01 

3.08 

3.15 

3.23    3.30 

3.38 

54 

37 

2.25 

2.29 

2.33 

2.36 

2.40 

2.44 

2,  49  2,  53  2.  58  2.  O:.:  2,  07  2,  ''J 

2:7s 

2.  84  2.  90 

2.95 

3.02' 

3.08 

3.15 

3.23 

3.30    3.38 

3.47i     63 

38 

2.30 

2.34 

2.38 

2.42 

2. 46  2. 50 

2,55  2,59  2.64,2.69 

2.74i2.79 

2,85 

2.  90  2,  96 

3,02 

3.09: 

3,16 

3.23 

3.30 

3.38    3.46 

3.551     52 

39 

2.35 

2.39 

2.43 

2.47 

2.512,56 

2.60  2.65  2.70  2.75 

2.  80 '2. 85 

2,91 

2.  97  3,  03 

.3.09 

3.161 

3,23 

3.30 

3.37 

3.45    3.53 

3. 62     51 

40 

2.40 

2.44 

2.48 

2.52 

2.  57  2.  61 

2.66  2.70  2.75  2.80 

2.  86,, 2,  91 

3.97 

3.03 

3,09 

3.16 

3.22 

3.29 

3.37 

3.45 

3.53    3.61 

3.  70j    50 

41 

2.45 

2.49 

2.53 

2.58 

2.  62  2.  66 

2.7l'2.76  2.812.86 

2,92:2,97 

3.03 

3.09 

3.16 

3,22 

3,29 

3.36:  3.44 

3.52 

3.60    3.69 

3.78 

40 

42 

2.50 

2.54 

2.58 

2.63 

2.  67  2.  72 

2.  77  2. 81  2.  87|2.  92  2.  97, 

3.03 

3.09 

3.15 

3,22 

3,29 

3.36 

3.43;  3.51 

3.59 

3.  67 1  3.  76 

3.85 

48 

43 

2.55 

2.59 

2.63 

2.68 

2.  72  2.  77 

2.82  2.87,2.922.98  3,03 

3.09 

3,15 

3.21 

3.28 

3.35 

3.42 

3.50]  3.57 

3.66 

3.74    3.83 

3.93 

47 

44 

2.60 

2.64 

2.68 

2.73 

2.  77  2.  82 

2.87,2.92  2.98,3.03  3.09 

3.15 

3.21 

3.27 

3.34 

3.41 

3.48 

3.56:  3.64 

3.72 

3.81   3.91 

4.00 

46 

45 

2.65 

2.69 

2.73 

2.78 

2.  82  2.  87 

2. 92|2. 97  3. 03  3.  08  3. 14 

8.20 

3.27 

3.33 

3.40 

3.47 

3.55 

3.62  3.71 

3.79  3.88   3.97| 

4.07 

45 

224 


A  ma:n^ual  of  topogeaphic  methods. 


Table  XXVIII. — Factors  for  reduction  of  transit  ohservations — Continued. 
Azimuth  factor  A=sm  C  sec.  5.    Star's  decliuation  ±  6.    Inclmation  factor  B  =  cos  i  sec.  6. 


460 

74J= 

2. 6n 

74r 

75° 

75io 

75J= 
2.S7 

75r 
2.92 

76° 
2.97 

76i0  76io 
3.  03  3.  OS 

76J° 

770 

77i° 
3.20 

77J° 
3.32 

77i° 
3.39 

78° 
3.46 

78i° 
3.53 

78J° 
3.01 

78i° 
3.69 

79° 

79i° 
3.86 

79J° 
3.95 

79J° 
4.04 

80° 
4.14 

i 

2.7s'2.S2 

3. 14'3.  20 

3.77 

44° 

47 

^   7 

■J.  1-17 

_\  |i'^ 

■J,  117 

11.  li"J 

:i,  08  :i.  i: 

11  IP  11,  2,"  11  "1 

11,  118 

11,  4," 

3.52'  3.59 

3.  67i  3.75 

3.83 

3.92 

4.01 

4.11 

4.21 

43 

48 

11, '57 

3.05 

3.73 

3.81 

3.89 

3.98 

4.08 

4.18 

4.28 

42 

49 

,1,03 

3.71 

3.79 

3.87 

3.96 

4.05 

4.14 

4.24 

4.35 

41 

50 

J  ;il 

"'■'-'' 

■■!"■ 

.1.  11 

.1,1: 

■■■  ■-'  ■'■  -;:" 

11, 114. 1,41    :    ,:; 

1,  -■! 

1   ill 

11,  08 

3.76 

3.84 

3.93 

4.02 

4.11 

4.20 

4.30 

4.41 

40 

51 

2.91 

2.95 

3.00 

3.05 

3.10 

3.16 

3.21 

3. 27  3. 33 

3.39 

3.45  11,5. 

11.00 

11.74 

3.82 

3.90 

3.98 

4.07 

4.17 

4.26 

4.37 

4.48 

39 

52 

2.95 

3.00 

3.04 

3.09 

3.15 

3.20 

3.26 

3.313.38 

3.44 

3.  .10  11,  ,-.7 

11.  114 

11,  71 

3.79 

3.87 

3.95 

4.04 

4.13 

4.22 

4.32 

4.43 

4.54 

38 

63 

2.99 

3.04 

3.09 

3.14 

3.19 

3.24 

3.30 

3.36  3.42 

3.48 

3.  35  3.  02 

11,  00 

3,  77 

11.84 

3.92 

4.01 

4.09 

4.19 

4.28 

4.3b 

4.49 

4.60 

37 

54 

3.03 

3.08 

3.13 

3.18 

3.23 

3.29 

3.34 

3.40  3.47 

3.53 

3.  60  3.  67 

3.74 

3.  81 

3.89 

3.97 

4.00 

4.15 

4.24 

4.34 

4.44 

4.55 

4.66 

36 

55 

3.07 

3.11 

3.16 

3.22 

3.27 

3.33 

3.39 

3.45  3.51 

3.57 

3.  64  3.  71 

3.78 

3.86 

8.94 

4.02 

4.11 

4.20 

4.29 

4.39 

4.50 

4.60 

4.72 

35 

56 

3.10 

3. 15 

3.20 

3.26 

3.31 

3.37 

3.  43 

3.  49  3.  55 

3,62 

3.  OS  3.  70 

3.  Sll 

3.91 

3.99 

4.07 

4.16 

4.25 

4.34 

4.44 

4.55 

4.66 

4.77 

•34 

57 

::.  11 

;  l:i 

1.  -11 

1,  17 

1,  Til  11  81' 

1,  8- 

11,  0,' 

4,04 

4.12 

4.21 

4.30 

4.39 

4.50 

4.60 

4.72 

4.83 

33 

58 

1.08 

4.16 

4.25 

4.35 

4.44 

4.55 

4.65 

4.77 

4.88 

32 

59 

>  ,11   ;  i  1 7 

4.12 

4.21 

4.30 

4.39 

4.49 

4.60 

4.70 

4.82 

4.94 

31 

(iO 

.:.-,! 

;,-h 

'■-i'' 

1  ,-.s 

1  <i.;:i  71 

I, .11 

.1, 

4,  17 

4.25 

4.34 

4.44 

4.54 

4.64 

4.75 

4.87 

4.99 

SO 

61 

3.27 

3.33 

3.38 

3.44 

3.49 

3.55 

3.62 

3.  683.  75 

3.  82 

3.^93.96 

4.04 

4.12 

4.21 

4.29 

4.39 

4.48 

4.58 

4.69 

4.80 

4.92 

5.04 

29 

62 

3.30 

3.36 

3.41 

3.47 

3.53 

3.59 

3.65 

3.  72  3.  78 

3!  85 

3.  92  4.  00 

4.08 

4.10 

4.25 

4.34 

4.43 

4.53 

4.63 

4.73 

4.85 

4.96 

5.08 

28 

63 

3.33 

3.39 

3.44 

3.50 

3.56 

3.62 

3.68 

3.  75  3.  82 

3.89 

3.  96  4.  04 

4.12 

4.20 

4.29 

4.38 

4.47 

4.57 

4.67 

4.78 

4.89 

5.01 

5.13 

27 

64 

3.36 

3.42 

3.47 

3.53 

3.59 

3.65 

3.72 

3.  78,3.  85 

3.92 

4.  00  4.  07 

4.15 

4.24 

4.32 

4.41 

4.51 

4.61 

4.71 

4.82 

4.93 

5.05 

5.18 

26 

65 

3.39 

3.45 

3.50 

3.56 

3.62 

3.68 

2.75 

3.  8113.  88 

3.95 

4.  03  4. 11 

4.19 

4.27 

4.36 

4.45 

4.55 

4.65 

4.75 

4.86 

4.97 

5.09 

5.22 

25 

66 

3,  J2 

:?.  47 

_ 

■5,  50 

l.fi.T 

3  71 

?,  7S 

!.  84  3,  91 

3,  99 

4.  004.  14 

4,22 

4.31 

4.40 

i.49 

4.58 

4.68 

4.79 

4.90 

5.01 

5.14 

5.26 

24 

67 

:.  4-1 

;   il- 

:    .  1 

;  ^i 

1    -T    :  04 

4,  I'L' 

1,  0:14,  17 

t,  20 

4,  1-14 

4.43 

4.52 

4.62 

4.72 

4.82 

4.94 

5.05 

5.18 

5.30 

23 

68 

1. 1 1  ■, 

1-  1;;  1,  20 

1,  2.^ 

4,  117 

4.46 

4.55 

4.65 

4.75 

4.86 

4.97 

5.09 

5.21 

5.34 

22 

69 

4,  11, 

4.  1,",  1,  2,1 

1.112 

4,40 

4.49 

4.58 

4.68 

4.79 

4.89 

5.00 

5.12 

5.25 

5.38 

21 

JO 

J.i: 

...  i; 

o.GJ 

;.GJ 

J.  t; 

,.j: 

..  Ml 

.1.  Li.j  4.  u;; 

4.  lu 

4.  18  4.25 

4.34 

4.43 

4.52 

4.61 

4.71 

4.82 

4.93 

5.04 

5.16 

5.28 

5.41 

20 

71 

3.54 

3.60 

3.65 

3.71 

3.78 

3.84 

3.91 

3.  98*4.  05 

4.13 

4.  20  4.  28 

4.37 

4.40 

4.55 

4.64 

4.74 

4.85 

4.96 

5.07 

5.19 

5.32 

5.45 

19 

72 

3.56 

3.63 

3.67 

3.74 

3.80 

3.86 

3.93 

4.  00  4.  07 

4.15 

4.  23!4.  31 

4.39 

4.48 

4.57 

4.67 

4.77 

4.88 

4.98 

5.10 

5.22 

5.34 

5.48 

18 

73 

3.58 

3.64 

3.69 

3.76 

3.82 

3.89 

3.95 

4.  02  4. 10 

4.17 

4.25:4.33 

4.42 

4.51 

4.60 

4.70 

4.80 

4.90 

5.01 

5.13 

5.25 

5.37 

5.51 

17 

74 

3.60 

3.65 

3.71 

3.78 

3.84 

3.91 

3.97 

4.  04;4. 12 

4.19 

4.  2714.  36 

4.44 

4  53 

4.62 

4.72 

4.82 

4.93 

5.04 

5.15 

5.27 

5.40 

5.53 

16 

75 

3.61 

3.67 

3.73 

3.79 

3.86 

3-92 

3.99 

4.06  4.14 

4.21 

4.  29  4.  38 

4.46 

4.55 

4;  65 

4.74 

4.84 

4.95 

5.06 

5.18 

5.30 

5.43 

5.56 

15 

76 

3.64 

3.  on 

'.  "-■ 

,  5, 

.94 

4.  01 

4.  OS  4.  ir. 

4,  2'_1 

4.  .11 '4.  40 

4,4s 

4.  .57 

4.07 

4.76 

4.87 

4.97 

5.09 

5.20 

5.32 

5.45 

5.59 

14 

77 

3.65 

3.711 

-.1 

'.  'Mi 

4.  03 

1.10  4,17 

1,  11114.41 

4.  ,'iO 

1,  ,'.0 

4.68 

4.78 

4.89 

4.90 

5.11 

5.22 

5.35 

5.47 

5.61 

13 

78 

3,66 

3.7: 

'.  07 

4.  04 

4.11  4.111 

4.  27 

4,  11,'.  4,411 

1,  ,'.2 

4,01 

4.70 

4.80 

4.9] 

5.01 

5.13 

5.24 

5.37 

5.50 

5.63 

12 

79 

3.67 

3.7.1 

1.119 

4.  OlJ 

4.1114.21 

4,  1;,^ 

4,  110  4.4,'. 

1.  ,'.4 

I,  OH 

4.72 

4.82 

4.92 

5.03 

5.14 

5.26 

5.39 

5.52 

5.65 

11 

SO 

3.68 

3.74 

i.sl 

i.bl 

0. 113 

4.00 

4.07 

4.144.22 

4.  3U 

4.  38  4.  40 

4.55 

4.04 

4.74 

4.84 

4.94 

5.05 

5.16 

5.28 

5.40 

5.54 

5.67 

10 

81 

3.70 

3.75 

3.82 

3.88 

3.94 

4.01 

4.08 

4.16  4.23 

4.31 

4.  39  4.  48 

4.  .56 

4.65 

4.75 

4.85 

4.95 

5.06 

5.18 

5.30 

5.42 

5.55 

5.69 

9 

82 

3.71 

3.76 

3.83 

^.89 

3.96 

4.02 

4.09 

4. 17  4.  24 

4.  32 

4.  40  4.  49 

4.57 

4.67 

4.76 

4.86 

4.97 

5.08 

5.19 

5.31 

5.43 

5.56 

5.70 

S 

83 

3.72 

3.77 

3.84 

3.90 

3.96 

4.03 

4.10 

4. 18  4,  i,". 

4.  414.  .io 

4.  :'.9 

4.  08 

4.78 

4.87 

4.98 

5.09 

5.20 

5.32 

5.45 

5.58 

5.72 

7 

84 

3.72 

3.78 

3.84 

3.91 

3.97 

4.04 

4.11 

4,  18  4,20 

4.34 

4.  42  4.  51 

4.00 

4.09 

4.79 

4.88 

4.99 

5.10 

5.21 

5.33 

5.46 

5.59 

5.73 

6 

85 

3.73 

3.79 

3.85 

3.91 

3.98 

4.05 

4.12  4.  1114  27 

4. 115 

4,4114.51 

4.60 

4.09 

4.79 

4.89 

5.00 

5.11 

5.22 

5.34 

5.47 

5.60 

5.74 

5 

86 

3.73 

3.711 

4.  ;i.' 

1,  l-jli.^M  L-27 

4.35 

4.43  4,52 

4.  01 

4.70 

4.80 

4.90 

5.00 

5.11 

5.23 

5.35 

5.47 

5.61 

5.74 

4 

87 

3.74 

3.71. 

4.  30  4.  44  4.  52 

4.02 

4.714.81 

4.90 

5.01 

5.12 

5.23 

5.35 

5.48 

5.61 

5.75 

3 

88 

3.74 

3,8' 

4,36  4.44  4.51! 

4.02 

4.  71  4.  81 

4.91 

5.01 

5.12 

5.24 

5.36 

5.48 

5.61 

5.75 

2 

89 

3.74 

S.Su 

1.  ,m1 

1  M,; 

.  ll'.i 

1.  ml 

1.  1:1  -I. -Jl  4.  1.8  4.30  4.44  4.53 

4.62 

4.  71  4.  81 

4.91 

5.01 

5.12 

5.24 

5.36 

5.49 

5.62 

5.76 

1 

90 

3.74 

3.80 

J.  Sli 

3!  93 

3.99 

4.06 

4. 1314.  21  4.  28,4.  36J4.  44l4.  53  4.  62 

4.  7114.  81 

4.91 

5.02 

5.13 

5.24 

5.36 

5.49 

5.62 

5.76 

0 

Table  XXIX. — For  reducing  observations  for  latitude  hy  Talcotfs  metliod. 

[Extracted  from  Appendix  14.    TJnited  States  Coast  and  Geodetic  Survey,  Eeport  for  1880.] 

Correction  for  differential  refraction. — The  difference  of  refraction  for  any  pair  of 
stars  ia  so  small  that  we  can  uegieet  the  variation  in  the  state  of  the  atmosphere  at  the 
time  of  the  observation  from  that  mean  state  supposed  in  the  refraction  tables.  The 
refraction  being  nearly  proportional  to  the  tangent  of  the  zenith-distance,  the  differ- 
ence of  refraction  for  the  two  stars  will  be  given  by — 
r— r'=57".7  sin  [z—z')  sec.-s; 

and  since  the  difference  of  zenith-distances  is  measured  by  the  micrometer,  the  follow- 
in  £;  table  of  correction  to  the  latitude  for  differential  refraction  has  been  prepared 


EEDUCTION  OF  LATITUDE  OBSEEVATIONS. 


225 


for  the  argument  ^difference  of  zenith-distance,  or  J  difference  of  micrometer-reading 
on  the  side,  and  the  argument  "Zenith-distance"  on  the  top.  The  sign  of  the  cor- 
rection is  the  same  as  that  of  the  micrometer  difference. 


i  diff.  in 

Zenith-distance 

i  dift-.  in 

zenith- 
distance. 

0° 

lOo 

20°- 

25° 

30° 

35° 

distance. 

0° 

10°     20° 

25° 

30° 

35° 

0 

.00 

.00 

.00 

.00 

.00 

.00 

6.5 

11 

.11 

K 

.13 

.14 

.16 

0.5 

.01 

.01 

.01 

.01 

.01 

.01 

7 

IX 

.12 

13 

.14 

.15 

.18 

1 

.02 

.02 

.02 

.02 

.02 

.02 

7.5 

13 

.13 

14 

.15 

.16 

.19 

1.5 

.02 

.03 

.03 

.03 

.03 

.03 

8 

13 

.14 

15 

.16 

.18 

.21 

2 

.03 

.03 

.04 

.04 

.04 

.05 

8.5 

14 

.15 

16 

.17 

.19 

.22 

2.5 

.04 

.04 

.05 

.  .05 

.05 

.06 

9 

15 

.16 

17 

.18 

.20 

.23 

3 

.05 

.05 

.06 

.06 

.07 

.08 

9.5 

IB 

.17 

18 

.20 

.21 

.24 

3.5 

.06 

.06 

.07 

.07 

.08 

.09 

10 

17 

.18 

19 

.21 

.23 

.26 

4 

.07 

.07 

.08 

.08 

.09 

.10 

10.5 

IK 

.19 

W 

.22 

.24 

.27 

4.5 

.08 

.08 

.09 

.09 

.10 

.11 

11 

IK 

.19 

21 

.23 

.25 

.28 

5 

.08 

.09 

.10 

.10 

.11 

.13 

11.5 

1» 

.20 

•>v. 

.24 

.26 

.30 

5.5 

.09 

.10 

.10 

.11 

.12 

.14 

12 

2() 

.21 

23 

.25 

.27 

.31 

6 

.10 

.10 

.11 

.12 

.13 

.15 

Reduction  to  the  meridian. — First,  when  the  line  of  collimation  of  the  telescope  is 
off  the  meridian,  the  instrument  having  been  revolved  in  azimuth  and  the  star  observed 
at  the  hour-angle  t,  near  the  middle  thread,  then 

2  sin^  hr    cos  a  cos  d 
m= — ^ — -—.  4--= — 

sm  1"  sm  C 

and  the  correction  to  the  latitude,  if  the  two  stars  are  observed  off  the  meridian 
=  ^  (m'—m).    The  value  of 

2  sin^  ^T 
t  sin  1" 

for  every  second  of  time  up  to  two  minutes  (a  star  being  rarely  observed  at  a  greater 
distance  than  this  from  the  meridian  in  zenith-telescope  observations'!  is  given  in  the 
following  table : 


- 

Term. 

- 

Term. 

^ 

Term. 

- 

Term. 

- 

Term. 

Term. 

1 

0.00 

21 

0.24 

41 

0.91 

61 

2.03 

s. 
81 

3.58 

101 

6.56 

'2 

0.00 

22 

0.26 

42 

0.96 

62 

2.10 

82 

3.67 

102 

5.67 

3 

0.00 

23 

0.28 

43 

1.01 

63 

2.16 

83 

3.76 

103 

6.78 

4 

0.01 

24 

0.31 

44 

1.06 

64 

2.23 

84 

3.85 

104 

5.90 

5 

0.01 

25 

0.34 

45 

1.10 

65 

2.31 

85 

3.94 

105 

6.01 

6 

0.02 

26 

0.37 

46 

1.15 

66 

2.38 

86 

4.03 

106 

6.13 

7 

0.02 

27 

0.40 

47 

1.20 

67 

2.45 

87 

4.12 

107 

6.24 

8 

0.03 

28 

0.43 

48 

.1.26 

■  68 

2.52 

88 

4.22 

108 

6.36 

9 

0.04 

29 

0.46 

49 

1.31 

69 

2.60 

89 

4.32 

109 

6.48 

10 

0.05 

30 

0.49 

50 

1.36 

70 

2.67 

90 

4.42 

110 

6.60 

11 

0.06 

31 

0.52 

51 

1.42 

71 

2.75 

91 

4.52 

111 

6.72 

12 

0.08 

32 

0.56 

52 

1.48 

72 

2.83 

92 

4.62 

112 

6.84 

13 

0.09 

33 

0.59 

53 

1.53 

73 

2.91 

93 

4.72 

113 

6.06 

14 

0.11 

34 

0.63 

54 

1.59 

74 

2.99 

94 

4:82 

114 

7.09 

15 

0.12 

35 

0.67 

65 

1.65 

75  » 

3.07 

95 

4.92 

115 

7.21 

16 

0.14 

36 

0.71 

56 

1.71 

76 

3.15 

90 

5.03 

116 

7.34 

17 

0.16 

37 

0.75 

57 

1.77 

77 

3.23 

97 

5.13 

117 

7.46 

18 

0.18 

38 

0.80 

58 

1.83 

78 

3.32 

98 

5.24 

118 

7.60 

.19 

0.20 

39 

0.83 

59 

1.89 

79 

3.40. 

99 

5.34 

119 

7.72 

20 

0.22 

40 

0.87 

60 

1.96 

80 

3.49 

100 

5.45 

120 

7.85 

MON  XXII- 


-15 


226 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Seco7ully,  when  the  star  is  observed  oft'  the  liue  of  coUimatioii,  the  instrumeut 
remaining  in  the  plane  of  the  meridian,  then 


m— — -. — zrvr-  sin  0  cos  o 
sm  1" 


2  sin^  iT_ 
sin  1" 


i  siu2(J 


and  the  correction  to  the  latitude  is  half  of  this  quantity,  whether  the  star  be  north  or 
south,  and  if  the  two  stars  forming  a  pair  are  observed  off  the  line  of  collimation,  two 
such  corrections,  separately  computed,  must  be  added  to  the  latitude.  If  the  stars 
should  be  south,  of  the  equator,  the  essential  sign  of  the  correction  is  negative.  The 
value  of  m  for  every. 5°  of  declination  is  given  in  the  following  table: 


IDs. 

15. 

205. 

25s. 

30s. 

35s. 

405. 

45s. 

50s. 

55s. 

60s. 

6 

„ 

„ 

„ 

„ 

„ 

„ 

„ 

„ 

„ 

S 

5° 

.00 

.01 

.02 

.03 

.04 

.06 

.08 

.10 

.12 

.14 

.17 

85° 

10 

.01 

.02 

.04 

.06 

.08 

.11 

.15 

.19 

.23 

.28 

.34 

80 

15 

.01 

-.03 

.05 

.09 

.12 

.17 

.22 

.28 

.34 

.41 

.49 

75 

20 

.02 

.04 

.07 

.11 

.16 

.22 

.28 

.36 

.44 

.53 

.63 

70 

25 

.02 

.05 

.08 

.13 

.19 

.26 

.34 

.42 

.52 

.63 

.75 

65 

30 

.02 

.05 

.09 

.15 

.21 

.29 

.38 

.48 

.59 

.71 

.85 

60 

35 

.03 

.06 

.10 

.16 

.23 

.31 

.41 

.53 

.64 

.77 

.92 

55 

40 

.03 

.06 

.11 

.17 

.24 

.33 

.43 

.54 

.67 

.81 

.97 

50 

43 

.03 

.06 

.11 

.17 

.25 

.33 

.44 

.55 

.68 

.82 

.98 

45 

Table  XXX. — For  facilitathifi  the  reduction  of  observatio7is,  on  close  circumpolar  stars^ 
made  in  determining  the  value  of  a  revolution  of  the  micrometer. 

[Extmcted  from  Appeudix  14.    TJ.  S.  Coast  and  Geodetic  Surve;Vi  Keport  for  1880.] 

Let  ?=difference  of  time  of  observation  and  elongation  of  the  star,  and  «"=num- 
ber  of  seconds  of  arc  in  the  direction  of  the  vertical  from  elongation,  then 


cos  S  sin  t 
sin  1" 


for  which  we  can  write 


"z=15coS(y]  *-i(L5sinl")'*= 


where  t  is  expressed  in  seconds  of  time.  It  is  convenient  to  apply  the  term  ^  (15  siu 
l")^^'  to  the  observed  time  of  noting,  additive  to  the  observed  time  before,  and  sub- 
tractive  after,  either  elongation.  The  following  table  gives  the  value  of  i(15  sin 
vyf,  also  of  the  additional  term 

—120  (15  sin  1")^  f  when  sensible,  for  every  minute  of  time  from  elongation  to  QS"". 


t     j  Term. 

1 
t     1  Term. 

1 

t 

Term. 

t 

Term. 

t 

Term. 

t 

Term. 

m 

J 

m. 

s. 

m 

s. 

m. 

«. 

m. 

s. 

m. 

s. 

6 

0.0 

16 

0.8 

26 

3.  .3 

36 

8.9 

46 

18.5 

56 

33.3 

0.1 

17 

0.9 

27 

3.7 

37 

9.6 

47 

19.7 

57 

35.1 

8 

0.1 

18 

1.1 

28 

-'4.2 

38 

10.  t 

48 

21.0 

58 

37.0 

9 

0.1 

19 

1.3 

29 

4.6 

39 

11.3 

49 

22.3 

59 

39.0 

10 

0.2 

20 

1.5 

30 

5.1 

40 

12.2 

50 

23.7 

60 

41.0 

11 

0.3 

21 

1.8 

31 

5.7 

41 

13.1 

51 

25.2 

61 

43.1 

12 

0.3 

22 

2.0 

■32 

6.2 

42 

14.1 

52 

26.7 

62 

45.2 

13 

0.4 

23 

2.3 

33 

6.8 

43 

15.1 

53 

28.3 

63 

47.4 

14 

0.5 

24 

2.6 

34 

7.5 

44 

16.2 

54 

29.9 

64 

49.7 

15 

0.6 

25 

3.0 

35 

8.2 

45 

17.3 

55 

31.6 

65 

52.1 

COirV^EESION  OF  SIDEEAL  INTO  MEAN  TIME. 


227 


Table  XXXI. — For  converting  intervals  of  sidej'eal  mto  coi'responding  intervals  of  mean  solar  time, 
[Extracted  from  Lee'a  Tables.] 


Hours. 

Minutes. 

Seconds. 

ft. 

m    s 

m 

s. 

m 

J 

J 

, 

J 

J 

1 

0  ' 09.  830 

1 

0.164 

31 

5.079 

i 

o.'6o3 

31 

0.085 

2 

0  19.  659 

2 

0.328 

32 

5.242 

2 

0.005 

32 

0.087 

3 

0  29.489 

3 

0.491 

33 

5.406 

3 

0.008 

33 

0.090 

4 

0  39. 318 

4 

0.655 

34 

5.570 

4 

0.011 

34 

0.093 

5 

0  49.148 

5 

0.819 

35 

5.734 

5 

0.014 

35 

0.096 

6 

0  58.977 

6 

0,983 

36 

5.89S 

6 

0.016 

36 

0.098 

7 

1  08. 807 

7 

1.147 

37 

6.062 

7 

0.019 

37 

0.101 

8 

1  18. 636 

8 

1.311 

38 

6.225 

8 

0.022 

38 

0.104 

9 

1  28.466 

9 

•1.474 

39 

6.389 

9 

0.025 

39 

0.106 

10 

1  38.296 

10 

1.638 

40 

6.553 

10 

0.027 

40 

0.109 

11 

1  48.125 

11 

1.802 

41 

6.717 

11 

0.030 

41 

0.112 

12 

1  57.955 

12 

1.966 

42 

6.881 

12 

0.033 

42 

0.115 

13 

2  07.784 

13 

2.130 

43 

7.044 

13 

0.036 

43 

0.118 

14 

2  17.614 

14 

2.  294 

44 

7.208 

14 

0.038 

44 

0.120 

15 

2  27.443 

15 

2.457 

45 

7.372 

15 

0.041 

45 

0.123 

16 

2  37.  273  , 

16 

2.621 

46 

7.536 

16 

0.044 

46 

0.126 

17 

2  47.103 

17 

2.785 

47 

7.700 

17 

0.047 

47 

0.128 

18 

2  56.932 

18 

2.949 

48 

7.864 

18 

0.049 

48 

0.131 

19 

3  06. 762 

19 

3.113 

49 

8.027 

19 

0.052 

49 

0.134 

20 

3  16.591 

20  ' 

3.277 

50 

8.191 

20 

0.055 

50 

0.137 

21 

3  26.421 

21 

3.440 

51 

8.355 

21 

0.057 

51 

0.140 

22 

3  36. 250 

22 

3.604 

52 

8.519 

22 

0.060 

52 

0.142 

23 

3  46.080 

23 

3.  768 

53 

8.083 

23 

0.063 

53 

0.145 

24 

3  55.909 

24 

3.932 

54 

8.847 

24 

0.066 

54 

0.148 

25 

4.096 

55 

9.010 

25 

0.068 

55 

0.150 

26 

4.259 

56 

9.174 

26 

0.071 

50 

0.153 

27 

4.423 

57 

9.338 

27 

0.074 

57 

0.156 

28 

4.587 

58 

9.502 

28 

0.076 

58 

0.159 

29 

4.751 

59 

9.666 

29 

0.079 

59 

0.161 

30 

4.915 

60 

9.830 

30 

0.082 

60 

0.164 

228. 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXII. — For  converting  intervals  of  mean  solar  time  into  corresponding  intervals  of  sidereal  time. 

[Extracted  from  Lee's  Tables.] 


Hours. 

Minutes. 

Seconds. 

h. 

m 

J 

m. 

c. 

m. 

s. 

». 

s. 

, 

s. 

1 

0 

09.  850 

1 

0.164 

31 

5'.092 

i 

0.'  003 

3i 

0.085 

0 

19.  713 

0.329 

32 

5.257 

2 

0.005 

32 

0.0S8 

3 

0 

29.  669 

3 

0.493 

33 

5.421 

3 

0.008 

33 

0.090 

4 

0 

39.426 

4 

0.657 

34 

5.585 

4 

0.011 

34 

0.093 

g 

0 

49.282 

5 

0.821 

35 

5.750 

5 

0.014 

35 

0.096 

6 

0 

59. 139 

6 

0.986 

36 

5.914 

6 

0.016 

36 

0.098 

7 

08.995 

7 

1.150 

37 

6.078 

7 

0.019 

37 

0.101 

8 

18.  852 

8 

1.314 

38 

6.242 

8 

0.022 

38 

0. 104 

0 

28. 708 

9 

1.478 

39 

6.407 

9. 

0.025 

39 

0.106 

10 

38.  565 

10 

1.643 

40 

6.571 

10 

0.027 

40 

0.109 

11 

48.421 

11 

1.807 

41 

6.735 

11 

0.030 

41 

0.112 

12 

58.278 

12 

1.971 

42 

6.900 

12 

0.033 

.42 

0.  116 

13 

US. 134 

13 

■  2. 136 

43 

7.064 

13 

0.036 

43 

0.118 

14 

2 

17.991 

14 

2.300 

44 

7.228 

14 

0.038 

44 

0.120 

16 

2 

27.  847 

15 

2.464 

45 

7.392 

15 

0.041 

45 

0.123 

16 

2 

37.  704 

16 

2.628 

46 

7.557 

16 

0.  044 

46 

0.126 

17 

2 

47.  560 

17 

2,793 

47 

7.721 

17 

0.047 

47 

0.129 

18 

2 

57. 416 

18 

2.957 

48 

7.885 

18 

0.049 

48 

0.131 

19 

3 

07.  273 

19 

3.121 

49 

8.050 

19 

0.052 

49 

0.134 

20 

3 

17. 129 

20 

3.285 

50 

8.214 

20 

0.055 

50 

0. 137 

21 

3 

26.  986 

21 

3.450 

51 

8.378 

21 

0.057 

51 

0.140 

22 

3 

36.  842 

22 

3.614 

52 

8.542 

22 

0.060 

52 

0.142 

23 

3 

46.  699 

23 

3.778 

53 

8.707 

23 

0.063 

53 

0.145 

24 

3 

56.555 

24 

3.943 

54 

8.871 

24 

0.066 

54 

0.148 

25 

4.107 

55 

9.  035 

25 

0.068 

5S 

0.151 

26 

4.271 

56 

9.199 

26 

0.071 

56 

0.153 

27 

4.436 

57 

9.364 

27 

0.074 

57 

0.156 

28 

4.600 

58 

9.528 

28 

0.077 

58 

0.159 

29 

4.764 

59 

9.692 

29 

0.079 

59 

0.161 

30 

4.928 

60 

9.856 

30 

0.082 

60 

0.164 

The  quantities  taken  from  this  table  mnat  be  added  to  a  i 
real  time. 


Qterval  to  obtain  the  correeponding  interval  in  side- 


CONYEESION  OF  AEG  INTO  TIME. 


•229 


Table   XXXIII. — To  comet-t  parts  of  the  equator  in  arc  into  sidereal  time,  or  to  convert  terrestrial  longitude 

in  arc  into  time. 

[Extracted  from  Lee's  Tables.]  * 


Degrees. 

De 

grees 

De 

grees 

De 

grees. 

Degrees. 

Degrees. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

1 

2 
3 
4 
5 

ft.  m. 

0  4 
0   8 
0  12 
0  16 
0  20 

61 
62 
63 
64 
65 

h. 

4 
4 
4 
4 
4 

m. 
4 
8 
12 
16 
20 

121 
122 
123 
124 
125 

ft. 
8 
8 

8 
8 

8 

4 
8 

12 
16 
20 

181 
182 
183 
181 
185 

ft.  m. 
12  4 

12   8 
12  12 
12  16 
12  20 

241 
242 
243 
244 
245 

ft.  -m. 
16   4 
16   8 
16  12 
16  16 
16  20 

301 
302 
303 
304 
305 

ft.  m. 
20   4 
20   8 
20  12 
20  16 
20  20 

6 
7 
8 
9 
10 

0  24 
0  28 
0  32 
0  36 
0  40 

66 
67 
68 
69 
70 

4 
4 
4 
4 
4 

24 
28 
32 
36 
40 

126 
127 
128 
129 
130 

8 
8 
8 
8 
8 

24 
28 
32 
36 
40 

186 
187 
188 
189 
190 

12  24 
12  28 
12  32 
12  36 
12  40 

246 
247 
248 
249 
250 

16  24 
16  28 
16  32 
16  36 
16  40 

306 
307 
308 
309 
310 

20  24 
20  28 
20  32 
20  36 
20  40 

11 
12 
13 
14 
15 

0  44 
0  48 
0  52 

0  56 

1  0 

71 
72 
73 
74 
75 

4 
4 
4 
4 
5 

44 
48 
.52 
56 
0 

131 
132 
133 
134 
135 

8 
8 
8 
8 
9 

44 
48 
52 
66 
0 

191 
192 
193 
194 
195 

12  44 
12  48 
12  52 

12  66 

13  0 

261 
252 
253 
254 
255 

16  44 
16  48 
16  52 

16  56 

17  0 

311 
312 
313 
314 
315 

20  44 
20  48 
20  62 

20  56 

21  0 

16 
17 
.   18 
19 
20 

1   4 
1   8 
1  12 
1  16 
1  20 

76 
77 
78 
79 
80 

.  5 
5 
5 
5 
5 

4 
8 
12 
16 
20 

136 
137 
138 
139 

140 

9 
9 
9 
9 
9 

4 
8 
12 
16 
20 

196 

197 
198 
199 
200 

13   4 
13   8 
13  12 
13  16 
13  20 

256 
257 
258 
259 
260 

17   4 
17   8 
17  12 
17  16 
17  20 

316 
317 
318 
319 
320 

21   4 
21   8 
21  12 
21  16 
21  20 

21 
22 
23 
24 
25 

1  24 
1  28 
1  32 
1  36 
1  40 

81 
82 
83 
84 
85 

5 
5 
5 
5 
5 

24 
28 
32 
36 
40 

141 
142 
143 
144 
145 

9 
9 
9 
9 
9 

24 
28 
32 
36 
40 

201 
202 
203 
204 
205 

13  21 
13  28 
13  32 
13  36 
13  40 

261 
262 
263 
264 
265 

17  24 
17  28 
17  32 
17  36 
17  40 

321 
323 
323 
324 
325 

21  24 
21  28 
21  32 
21  36 
21  40 

26 
27 
28 
29 
30 

1  4-1 
1  48 
1  52 

1  56 

2  0 

86 
87 
88 
89 
90 

5 

5 
5 
5 
6 

44 
48 
52 
56 
0 

146 
147 
148 
149 
150 

9 
9 
9 
9 
10 

44 
48 
52 
56 
0 

206 
207 
208 
209 
210 

13  44 
13  48 
13  52 

13  56 

14  0 

266 
267 
268 
269 
270 

17  44 
17  48  ■ 
17  ,52 

17  56 

18  0 

326 
327 
328 
329 
330 

21  44 
21  48 
21  52 

21  66 

22  0 

31 
32 
33 
34 
35 

2   4 
2   8 
2  12 
2  16 
2  20 

91 
92 
93 
94 
95 

6 
6 
6 
6 
6 

4 
8 
12 
16 
20 

151 

152 
153 
154 
155 

10 
10 
10 
10 
10 

4 
8 
12 
16 
20 

211 
212 
213 
214 
216 

14   4 
14   8 
14  12 
14  16 
14  20 

271 
272 
273 
274 
275 

18   4 
18   8 
18  12 
18  16 
18  20 

331 
332 
333 
334 
335 

22   4 
22   8 
22  12 
22  16 
22  20 

36 
37 
38 
39 
40 

2  24 
2  28 
2  32 
2  36 
2  40 

96 
97 
98 
99 

loo " 

6 
6 
6 
6 
6 

24 
28 
32 
36 
40 

156 
157 
168 
159 
160 

10 
10 
10 
10 
10 

24 
28 
32 
36 
40 

216 
217 
218 
219 
220 

14  24 
14  28 
14  32 
14  36 
14  40 

276 
277 
278 
279 
280 

18  24 
18  28 
18  32 
18  36 
18  40 

336 
337 
338 
339 
340 

22  24 
22  28 
22  32 
22  36 
22  40 

41 
42 
43 
44 
45 

2  44 
2  48 
2  52 

2  66 

3  0 

lOl 
l02 
l03 
l04 
l05 

6 
6 
6 
6 

7 

44 
48 
52 
56 
0 

161 
162 
163 
164 
165 

10 
10 
10 
10 
11 

44 
48 
52 
56 
0 

221 
222 
223 
224 
225 

14  44 
14  48 
14  52 

14  56 

15  0 

281 
282 
283 
284 
285 

18  44 
18  48 
18  62 

18  56 

19  0 

341 
342 
343 
344 
345 

22  44 
22  48 
22  52 

22  56 

23  0 

46 
47 
48 
49 
50 

3   4 
3   8 
3  12 
3  16 
3  20 

l06 
107 
108  . 
109 

no 

7 
7 
7 
7 
7 

4 
8 
12 
■6 
20 

166 
167 
168 
169 
170 

11 
11 
11 
11 
11 

4 
8 
12 
16 
20 

226 
227 
228 
229 
230 

15   4 
15   8 
15  12 
15  16 
15  20 

286 
287 
288 
•289 
290 

19   4 
19   8 
19  12 
19  16 
19  20- 

346 
347 
348 
349 
350 

23   4 
23   8 
23  12 
23  16 
23  20 

51 
52 
53 
54 
55 

3  24 
3  28 
3  32 
3  36 
3  40 

111 
112 
113 
114 
115 

7 
7 
7 
7 
7 

24 
28 
32 
36 
40 

171 
172 
173 
174 
175 

11 
11 
11 
11 
11 

24 
28 
32 
36 
40 

231 
232 
233 
234 
235 

15  24 
15  28 
15  32 
15  36 
15  40 

291 
292 
293 
294 
295 

19  24 
19  28 
19  32 
19  36 
19  40 

351 
352 
363 
354 
355 

23  24 
23  28 
23  32 
23  36 
23  40 

56 
57 
58 
59 
60 

3  44 
3  48 
3  52 

3  56 

4  0 

116 
117 
118 
119 
120 

7 
7 
7 

8 

44 
48 
52 
56 
0 

176 
177 
178 
179 
180 

11 
11 
11 
11 
12 

44 
48 
52 
56 
0 

236 
237 
238 
239 
240 

15  44 
15  48 
15  62 

15  56 

16  0 

296 
297 
398 
299 
300 

19  44 
19  48 
19  62 

19  56 

20  0 

356 
367 
358 
369 
360 

23  44 
33  48 
28  52 

23  66 

24  0 

230 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Taule  XXXIII. — To  coni'ert  parts  of  the  equator  in  arc  into  sidereal  time,  or  to  eonvert  teirestrial  longitude 
in  arc  into  time — Continued. 

[Extracted  from  Lee's  Tables.] 


1         Minutes. 

Minutes. 

Minutes. 

Seconds. 

Seconds. 

Seconds. 

Arc* 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

m     s 

, 

m     s. 

m.     s. 

„ 

J 

J 

1         1 

0      4 

21 

1    24 

41 

2    44 

1 

0.067 

21 

1.400 

41 

2.733 

(          2 

0      8 

22 

1    28 

42 

2    48 

2 

0.133 

22 

1.467 

42 

2.800 

3 

0    12 

23 

1    32 

43 

2    52 

3 

0.200 

23 

1.533 

43 

2.867 

* 

0    16 

24 

1    86 

44 

2    56 

4 

0.267 

24 

1.600 

44 

2.933 

5 

0    20 

25 

1    40 

45 

3      0 

5 

0.333 

25 

1.667 

45 

3.000 

6 

0    24 

26 

1    44 

,       46           3      4 

6 

0.400 

26 

1.733 

46 

3.067 

7 

0    28 

27 

1    48 

47     1      3      8 

•    7 

0.467 

27 

1.800 

47 

3.133 

8 

0    32 

28 

1    52 

48     ;      B    12 

8 

n.533 

28 

1.867 

48 

3.200 

9    1      0    36 

29 

1    56 

49     1      3    16 

9 

0.600 

29 

1.933 

49 

3.267 

10    I      0    40 

30 

2      0 

50           3    20 

10 

0.667 

30 

2.000 

50 

3.333 

11 

0    44 

31 

2      4 

51           3    24 

11 

0.733 

31 

2.067 

51 

3.400 

12 

0-   48 

32 

2      8 

52           3    28 

12 

0.800 

32 

2.133 

52 

3.467 

13 

0    52 

33 

2    12 

53           3    32 

13 

0.867 

33 

2.200 

53 

3.  533 

14 

0    56 

34 

2    16 

54           3     36 

14 

0.933 

34 

2.267 

54 

3.600    1 

15 

1      0 

35 

2    20 

55            3    40 

15 

1.000 

35 

2.333 

55 

3.667    < 

16     ,       1       4 

36 

2    24 

56           3     44 

16 

1.067 

36 

2.400 

56 

3.  733 

17     1       1       8 

37           2    28 

57            3    48 

17 

1.133 

37 

2.467 

57 

3.  800 

18            1     12 

38           2    32 

58            3     52     ' 

18 

1.200 

38 

2.633 

58 

3.867 

19            1     16 

39           2    36 

59     ,       3     56 

19 

1.267 

39 

2.600 

59 

3.933    1 

20            1     20 

40           2    40 

60    1      4      0 

20 

1.333 

40 

2.667 

60 

4.000    1 

1 

T.\Bi.E  XXXIV. — To  convert  sidereal  time  into  parts  of  the  equator  in  arc,  or  to  convert  time  into  terrestrial 

longitude  in  arc. 

[Extracted  from  Lee'.^  Tables.] 


Hours. 

Minutes. 

Seconds. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

h. 

0 

m 

o     - 

m 

o     , 

s. 

s. 

,  // 

1 

15 

1 

0  15 

31 

7  45 

1 

0  15 

31 

7  45 

2 

30 

2 

b  30 

32 

8  00 

2 

0  30 

32 

8  00 

3 

45 

3 

0  45 

33 

8  15 

3 

0  45 

33 

8  15 

4 

60 

4 

1  00 

34 

8  30 

4 

1  00 

34 

8  30 

5 

75 

5 

1  15 

35 

8  45 

5 

1  15 

35 

8  45 

6 

90 

6 

1  30 

36 

9  00 

6 

1  30 

36 

9  00 

7 

105 

7 

1  45 

37 

9  15 

7 

1  45 

37 

9  15 

8 

120 

8 

2  00 

38 

9  30 

8 

2  00 

38   " 

9  30 

9 

135 

9 

2  15 

39 

9  45 

9 

2  15 

39 

9  45 

10 

150 

10 

2  30 

40 

10  00 

10 

2  30 

40 

10  00 

11 

165 

11 

2  46 

41 

10  15 

11 

2  45 

41 

10  15 

12 

180 

12 

3  00 

42 

10  30 

12 

3  00 

42 

10  30 

13 

195 

13 

3  15 

43 

10  45 

13 

3  15 

43 

10  45 

14 

210 

14 

3  30 

44 

11  00 

14 

3  30 

44 

11  00 

15 

225 

.15 

3  45 

45 

11  15 

15 

3  45 

45 

11  15 

16 

240 

16 

4  00 

46 

11  30 

16 

4  00 

46 

11  30 

17 

255 

17 

4  15 

47 

11  45 

17 

4  15 

47 

11  45 

18 

270 

18 

4  30 

48 

12  00 

18 

4  30 

48 

12  00 

19 

285 

19 

4  45 

49 

12  15 

19 

4  45 

49 

12  15 

1      20 

300 

20 

5  00 

50 

12  30 

20 

5  00 

50 

12  30 

21 

315 

21 

5  15 

51 

12.45 

21 

5  15 

51 

12  45 

22 

330 

22 

5  30 

52 

13  00 

22 

5  30 

52 

13  00 

23 

345 

23 

5  45 

53 

13  15 

23 

5  45 

63 

13  15 

24 

360 

24 

6  00 

54- 

13  30 

24 

6  00 

54 

13  30 

25 

6  15 

55 

13  45 

25 

6  15 

55 

13  45 

26 

6  30 

56 

14  00 

26 

6  30 

56 

14  00 

27 

6  45 

57 

14  15 

27 

6  45 

57 

14  15 

28 

7  00 

58 

14  30 

28 

7  00 

58 

14  30 

29 

7  15 

59 

14  45 

29 

7  15 

59 

14  45 

30 

7  30 

60 

15  00, 

30 

7  30 

60 

15  00 

CONVERSION  OF  TIME  INTO  AEG.  231 

Table  XXXIV.— To  covrert  sidereal  lime  into  j)«)'(s  of  the  equator  in  arc,  eic,— Coutinued. 
[Extracted  from  Lee's  Tables.] 


Tenths  o: 

seconds 

Thou- 
sandths 

of  sec- 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

Time. 

Arc. 

onds  of 
time. 

s. 

0.21 

3.15 

s. 
0.41 

6.15 

s. 
0.61 

9.15 

0.81 

12. 15 

0.061 

0.015 

0.30 

0.22 

3.30 

0.42 

6,30 

0.62 

9.30 

0.82 

12.30 

0.002 

0.03 

0.45 

0.23 

3.45 

0.43 

6.45 

0.63 

9.45 

0.83 

12.45 

0.60 

0.24 

3.60 

0.44 

6.60 

0.64 

9.60 

0.84 

0.004 

0.05 

0.75 

0.25 

3.75 

0.45 

6.75 

0.65 

9.75 

0.85 

12.75 

0.005 

0.075 

0.90 

0.26 

3,90 

0.40 

6.90 

0.66 

9.90 

0.86 

12.90 

0.006 

0.090 

0.07 

1.05 

0.27 

4.U5 

0.47 

7.05 

0.67 

10.05 

0.87 

13.05 

0.007 

0.105 

0.08 

1.20 

0.28 

4.20 

0.48 

7.20 

0.68 

10.20 

0.88 

13.20 

0.008 

0,120 

1.35 

0.29 

4.35 

0.49 

7.35 

0.69 

10.35 

0.89 

0.009 

0.10 

1.50 

0.30 

4.50 

0.50 

7.50 

0.70 

10.50 

0.90 

13.50 

0,010 

0,150 

0.11 

1.65 

0.31 

4.65 

0.51 

7.65 

0.71 

10.65 

0.91 

13.65 

1.80 

0.32 

4.80 

0.52 

7.80 

0.72 

10. 80 

0.92 

13.80 

0.13 

1.95 

0.33 

4.95 

0.53 

7.95 

0.73 

10.95 

0.93 

13.95 

2.10 

0.34 

5.10 

0.54 

8.10 

0.74 

11.10 

0.94 

0.15 

2.25 

0.35 

5.25 

0.55 

8.25 

0.75 

11.25 

0.95 

14.25 

0.16 

2.40 

0.36 

5.40 

0.56 

8.40 

0.76* 

11:40 

0.96 

14.40 

0.17 

2.55 

0.37 

5.55 

0.57 

8.55 

0.77 

11.65 

0.97 

14.55 

0.18 

2.70 

0.38 

5.70 

0.58 

8.70 

0.78 

11.70 

0.98 

2.85 

0.39 

5.85 

0.59 

8.85 

0.70 

11.85 

0.99 

0.20 

3.00 

0.40 

6.00 

0.60 

9.00 

0.80 

12.00 

1.00 

15.00 

Table  XXXV. — Containiiuj  logarithms  of  nnmliers  from  1  to  11,000. 
[Extracted  froin  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

0 

_ 

20 

1. 30  103 

40 

L60  206 

60 

1.  77  815 

80 

1.90  309 

1 

0.00  000 

21 

1. 32  222 

41 

1.61  278 

61 

1.  78  533 

81 

1.  90  849 

2 

0.30  103 

22 

1. 34  242 

42 

1.  62  325 

62 

1.  79  239 

82 

1.  91  381 

3 

0. 47  712 

23 

1. 36  173 

43 

1.63  347 

63 

1.  79  934 

83 

1.  91  908 

4 

0. 60  206 

24 

1.  38  021 

44 

1. 64  345 

64 

1.80  618 

84 

1.92  428 

5 

0.  69  897 

25 

1.  39  79'J 

45 

L65  321 

65 

1. 81  291 

85 

1. 92  942 

6 

0.  77  815 

26 

L41  497 

46 

1. 66  276 

66 

1.81  954 

86 

1. 93  450 

7 

0.84  510 

27 

1.43  130 

47 

1.67  210 

67 

1.  82-  607 

87 

8 

0.90  309 

28 

1.44  716 

48 

1.  68  124 

68 

1.  83  251 

88 

1.94  448 

9 

0.  95  424 

29 

1.46  240 

49 

1.69  020 

69 

1.83  885 

89 

1.  94  939 

10 

1.  00  000 

30 

\.  47  712 

50 

1,  69  897 

70 

1.  84  510 

90 

1.  95  424 

11 

1.04  139 

31 

1.49  136 

51 

1.  70  757 

71 

1.85  126 

91 

1.  95  004 

12 

1.07  918 

32 

1.50  515 

52 

1.71  600 

72 

1.85  733 

.  92 

1.96  379 

13 

1.11  394 

33 

1,  51  851 

53 

1.  72  428 

73 

L86  332 

93 

1.  96  848 

14 

1. 14  613 

34 

1.53  148 

54 

1. 73  239 

74 

1.86  923 

15 

L17  609 

35 

1. 54  407 

55 

1.  74  036 

75 

L87  506 

95 

1.  97  772 

16 

1.20  412 

36 

1.  55  630 

56 

1.74  819 

76 

1.88  08 L 

96 

1. 98  227 

17 

1. 23  045 

37 

1.  56  820 

57 

1.75  587 

77 

1.88  649 

18 

1.  25  527 

33 

1.  57  978 

58 

1. 76  343 

78 

1.  89  209 

98 

1.99  123 

1.  27  875 

39 

1.59  106 

59 

1.  77  085 

79 

1. 89  763 

99 

1. 99  564 

20 

1. 30  103 

40 

1.  60  200 

60 

1.  77  815 

80 

1. 90  309 

100 

2.  00  000 

232 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXV. — Containing  Jogariihms  of  «itw6e7's  from  1  to  11,000- 
[Extracted  trom  GauSs'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

00  000 

30  103 

47  712 

60  206 

69  897 

77  815 

84  510 

90  309 

95  424 

1 

00  000 

04  139 

07  918 

11  394 

14  613 

17  609 

20  412 

23  045 

25  527 

27  875 

2 

30  103 

32  222 

34  242 

36  173 

38  021 

39  794 

41  497 

43  136 

44  716 

46  240 

3 

47  712 

49  136 

50  515 

51  851 

53  148 

54  407 

55  630 

56  820 

57  978 

59  106 

4 

60  206 

61  278 

62  325 

63  347 

64  345 

65  321 

60  276 

67  210 

68  124 

69  020 

5 

69  897 

70  757 

71  600 

72  428 

73  239 

74  036 

74  819 

75  587 

70  343 

77  085 

6 

77  815 

78  533 

79  239 

79  934 

80  018 

81  291 

81  954 

82  607 

83  251 

83  885 

84  510 

85  126 

85  733 

86  332 

.86  923 

87  506 

88  081 

88  649 

89  209 

89  763 

8 

90  309 

90  849 

91  381 

91  908 

92  428 

92  942 

93  430 

93  952 

94  448 

94  939 

9 

95  424 

95  904 

96  379 

96  848 

97  313 

97  772 

98  227 

98  677 

99  123 

99  564 

10 

00  OOO 

00  432 

00  860 

01  284 

01  703 

02  119 

02  531 

02  938 

03  342 

03  743 

11 

04  139 

04  532 

04  922 

05  308 

05  690 

00  070 

06  446 

06  819 

07  188 

07  555 

12 

07  918 

08  279 

08  636 

08  991 

09  342 

09  691 

10  037 

10  380 

10  721 

11  039 

13 

11  394 

11  727 

12  057 

12  385 

12  710 

13  033 

13  354 

13  672 

13  983 

14  301 

14 

14  613 

14  925 

15  229 

15  534 

15  836 

10  137 

16  435 

16  732 

17  020 

17  319 

15 

17  609 

17  898 

18  184 

18  469 

18  752 

19  033 

19  312 

19  590 

19  866 

20  140 

16 

20  412 

20  683 

20  952 

21  219 

21  484 

21  748 

22  Oil 

22  272 

22  631 

22  789 

17 

23  045 

23  300 

23  553 

23  805 

24  055 

24  304 

24  551 

24  797 

25  042 

25  285 

18 

25  527 

25  768 

26  007 

26  245 

26  482 

26  717 

26  951 

27  184 

27  416 

27  646 

19 

27  875 

28  103 

28  330 

28  55(1 

28  780 

29  003 

29  226 

29  447 

29  667 

29  885 

20 

30  103 

30  320 

30  535 

30  750 

30  963 

31  175 

31  387 

31  597 

31  806 

32  015 

21  ■ 

32  222 

32  428 

32  634 

32.  838 

33  041 

33  244 

33  445 

33  646 

33  846 

34  044 

22 

34  242 

34  439 

34  035 

34  830 

35  025 

35  218 

35  411 

35  603 

35  793 

35  984 

23 

36  173 

36  361 

30  549 

36  736 

36  922 

37  107 

37  291 

37  175 

37  058 

37  840 

24 

38  021 

38  202 

38  382 

38  561 

38  739 

38  917 

39  094 

39  270 

39  446 

39  620 

25 

39  794 

39  907 

40  140 

40  312 

40  483 

40  654 

40  824 

40  993 

41  162 

41  330 

26 

41  497 

41  064 

41  830 

41  996 

42  160 

42  325 

42  488 

42  651 

42  813 

42  976 

27 

43  136 

43  297 

43  457 

43  616 

43  775 

43  933 

44  091 

44  248 

44  404 

44  660 

28 

•  44  716 

44  871 

45  025 

45  179 

45  332 

45  484 

45  637 

45  788 

45  939 

46  090 

29 

46  240 

46  389 

40  538 

46  687 

46  835 

46  982 

47  129 

47  276 

47  422 

47  667 

30 

47  712 

47  857 

48  001 

48  144 

48  287 

48  430 

48  572 

48  714 

48  855 

48  996 

31 

49  136 

49  276 

49  415 

49  554 

49  693 

49  831 

49  969 

50  106 

50  243 

50  379 

32 

50  515 

50  651 

50  786 

50  920 

51  055 

51  188 

51  322 

51  455 

51  587 

51  720 

33 

51  851 

51  983 

52  114 

52  244 

52  375 

52  504 

52  634 

52  763 

52  892 

53  020 

34 

53  148 

53  275 

53  403 

53  529 

53  656 

53  782 

53  908 

54  033 

54  158 

54  283 

35 

54  407 

54  531 

54  654 

54  777 

54  900 

55  023 

55  145 

55  267 

53  388 

55  509 

36 

55  630 

55  751 

55  871 

55  991 

56  110 

56  229 

56  348 

56  467 

60  585 

56  703 

87 

56  820 

56  937 

57  054 

57  171 

57  287 

67  403 

57  519 

57  634 

57  749 

57  864 

38 

57  978 

58  092 

58  206 

58  320 

58  433 

58  546 

58  059 

58  771 

58  883 

58  995 

39 

59  106 

59  218 

59  329 

59  439 

59  550 

59  660 

59  770 

59  879 

59  988 

60  097 

40 

60  206 

60  314 

60  423 

60  531 

60  638 

60  746 

60  853 

60  959 

61  066 

61  172 

41 

61  278 

61  384 

61  490 

61  595 

61  700 

61  805 

61  909 

62  014 

62  118 

62  221 

42 

62  325 

62  428 

62  531 

62  634 

62  737 

62  839 

62  941 

63  043 

63  144 

63  246 

43 

63  347 

63  448 

63  548 

63  649 

63  749 

63  849 

63  949 

64  048 

64  147 

64  246. 

44 

64  345 

64  444 

64  542 

64  640 

64  738 

64  836 

64  933 

65  031 

65  128 

65  225 

45 

65  321 

65  418 

65  514 

65  610 

65  706 

65  801 

65  896 

65  992 

60  087 

66  181 

46 

66  276 

66  370 

66  464 

66  558 

66  652 

66  745 

66  839 

66  932 

67  025 

67  117 

47 

67  210 

67  302 

67  394 

67  486 

67  578 

67  069 

67  701 

67  852 

67  943 

68  034 

48 

68  124 

68  215 

68  305 

68  395 

68  485 

68  574 

68  664 

68  753 

68  842 

68  931 

49 

69  020 

69  108 

69  197 

69  285 

69  373 

09  461 

69  548 

69  636 

69  723 

69  810 

50 

69  897 

69  SJ84 

70  070 

70  157 

70  243 

70  329 

70  415 

70  501 

70  586 

70  672 

N". 

L.  0 

1 

2 

3 

4 

5, 

6 

7 

8 

9 

LOGARITHMS  OF  NUMBEES. 


233 


Table  XXXV. — Containing_  logarithms  of  numhers  from  1  to  11,000 — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

•  1 

2 

3 

4 

5 

6 

7 

8 

9 

60 

69  897 

69  984 

70  070 

70  157 

70  243 

70  329 

70  415 

70  501 

70  586 

70  672 

51 

70  757 

70  842 

70  927 

71  012 

71  096 

71  181 

71  265 

71  349 

71  433 

71  517 

52 

71  600 

71  684 

71  767 

71  850 

71  933 

72  016 

72  099 

72  181 

72  263 

72  346 

53 

72  428 

72  509 

72  591 

72  673 

72  754 

72  835 

72  916 

72  997 

73  078 

73  159 

54 

73  239 

73  320 

73  400 

73  480 

73  560 

73  640 

73  719 

73  799 

73  878 

73  957 

55 

74  036 

74  115 

74  194 

74  273 

74  351 

74  429 

74  607 

74  586 

74  663 

74  741 

56 

74  819 

74  896 

74  974 

75  051 

75  128 

75  205 

75  282 

75  358 

75  436 

75  511 

67 

.  75  587 

75  004 

75  740 

75  815 

75  891 

75  967 

76  042 

76  118 

76  193 

76  268 

58 

76  343 

76  418 

76  492 

76  567 

76  641 

76  716 

76  790 

76  864 

76  938 

77  012 

59 

77  085 

77  159 

77  232 

77  305 

77  379 

77  452 

77  525 

77  597 

77  670 

77  743 

60 

77  815 

77  887 

77  960 

78  032 

78  104 

78  176 

78  247 

78  319 

78  390 

78  462 

61 

78  533 

78  604 

78  675 

78  746 

78  817 

78  888 

78  958 

79  029 

79  099 

79  169 

62 

79  239 

79  309 

79  379 

79  449 

79  518 

79  588 

79  667 

79  727 

79  796 

79  865 

63 

79  934 

80  003 

80  072 

80  140 

80  209 

80  277 

80  346 

80  414 

80  482 

80  650 

64 

80  618 

80  686 

80  754 

80  82X 

80  889 

80  956 

81  023 

81  090 

81  158 

81  224 

65 

81  291 

81  358 

81  425 

81  491 

81  558 

81  624 

81  690 

81  757 

81  823 

81  889 

66 

81  954 

82  020 

82  086 

82  151 

82  217 

82  282 

82  347 

82  413 

82  478 

82  543 

67 

82  607 

82  672 

82  737 

82  802 

82  866 

82  930 

82  995 

83  059 

83  123 

83  187 

68 

83  251 

83  315 

83  378 

83  442 

83  506 

83  569 

83  632 

83  696 

83  759 

83  833 

69 

83  885 

83  948 

84  Oil 

84  073 

84  136 

84  198 

84  261 

84  323 

84  386 

84  448 

70 

84  510 

84  572 

84  634 

84  696 

84  757 

84  819 

84  880 

84  942 

85  003 

85  065 

71 

85  126 

85  187 

85  248 

85  309 

85  370 

85  431 

85  491 

85  552 

85  612 

85  673 

72 

85  733 

85  794 

85  854 

85  914 

85  974 

86  034 

86  094 

86  153 

86  213 

86  273 

73 

86  332 

86  392 

86  451 

86  510 

86  570 

86  629 

86  688 

86  747 

86  806 

86  864 

74 

86  923 

86  982 

87  040 

87  099 

87  157 

87  216 

87  274 

87  332 

87  390 

87  448 

75 

87  506 

87  564 

87  622 

87  679 

87  737 

87  795 

87  852 

87  910 

87  967 

83  024 

76 

8<  081 

88  138 

88  195 

88  252 

88  309 

88  366 

88  423 

88  480 

88  636 

88  593 

77 

88  649 

88  705 

88  762 

88  818 

88  874 

88  930 

88  986 

89  042 

89  098 

89  154 

78 

89  209 

89  205 

89  321 

89  376 

89  432 

89  487 

89  542 

89  597 

89  653 

89  708 

79 

89  763 

89  818 

89  873 

89  927 

89  982 

90  037 

90  091 

90  146 

90  200 

90  256 

80 

90  309 

90  363 

90  417 

90  472 

90  526 

90  580 

90  634 

90  687 

90  741 

90  795 

81 

90  849 

90  902 

90  956 

91  009 

91  062 

91  116 

91  169 

91  222 

91  276 

91  328 

83 

91  381 

91  434 

91  487 

91  540 

91  593 

91  645 

91  698 

91  751 

91  803 

91  855 

S3 

91  908 

91  960 

92  012 

92  065 

92  117 

92  169 

92  221 

92  273 

92  324 

93  376 

84 

92  428 

92  480 

92  531 

92  583 

92  634 

92  686 

92  737 

92  788 

92  840 

92  891 

85 

92  942 

92  993 

93  044 

93  095 

93  146 

93  197 

93  247 

93  298 

93  349 

93  399 

86 

93  450 

93  500 

93  551 

93  601 

93  651 

93  702 

93  752 

93  802 

93  852 

93  902 

87 

93  952 

94  002 

94  052 

94  101 

94  151 

94  201 

94  250 

94  300 

94  349 

94  399 

88 

94  448 

94  498 

94  547 

94  596 

94  645 

94  694 

94  743 

94  792 

94  841 

94  890 

89 

94  939 

94  988 

95  036 

95  085 

95  134 

95  182 

95  231 

95  279 

95  328 

95  376 

90 

95  424 

95  472 

95  521 

95  569 

95  617 

95  665 

95  713 

95  761 

96  809 

95  856 

91 

95  904 

95  952 

95  999 

96  047 

96  095 

96  142 

96  190 

96  237 

■  96  284 

96  333 

92 

96  379 

96  426 

96  473 

96  520 

96  567 

96  614 

96  661 

96  708 

96  755 

96  802 

93 

96  848 

96  895 

96  942 

96  088 

97  035 

97  081 

97  128 

97  174 

97  230 

97  267 

94 

97  313 

97  359 

97  405 

97  451 

97  497 

97  643 

97  589 

97  035 

97  681 

97  727 

95 

97  772 

97  818 

97  864 

97  909 

97  955 

98  000 

98  046 

98  091 

98  137 

98  182 

96 

98  227 

98  272 

98  318 

98  363 

98  408 

98  453 

98  498 

98  543 

98  588 

98  632 

97 

98  677 

98  722 

98  767 

98  811 

98  856 

98  900 

98  945 

98  989 

99  034 

99  078 

98 

99  123 

99  167 

99  211 

99  255 

99  300 

99  344 

99  388 

99  432 

99  476 

99  520 

99 

99  564 

99  607 

99  651 

99  695 

99  739 

99  782 

99  826 

99  870 

99  913 

99  957 

100 

00  000 

00  043 

00  087 

00  130 

00  173 

00  217 

00  260 

00  303 

00  346 

00  389 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

234 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXY. — Containing  Jogariihms  of  numbers  from  1  to  11,000 — Continued. 
[Extracted  from  Ciauss'  Logaritluuic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P. 

100 

00  000 

043 

087 

130 

173 

217 

260 

303 

346 

389 

101 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 

44 

43 

42 

102 

860 

903 

945 

988 

,030 

,072 

,115 

,157 

,199 

,242 

1 

4,4 

4,3 

4,2 

1.13 

01  284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

2 

8,8 

8,6 

8,4 

104 

703 

745 

787 

828 

870 

912 

953 

995 

,036 

.078 

3 

13,2 

12,9 

12'6 

105 

02  119 

100 

202 

243 

284 

325 

366 

407 

449 

490 

4 

17,6 

17,2 

16,8 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

808 

5 

22,0 

21,5 

21,0 

107 

938 

979 

,019 

,060 

,100 

,141 

.181 

,222 

,262 

,302 

6 

26,4 

25,8 

25,2 

108 

03  342 

383 

423 

463 

503 

543 

583 

623 

603 

703 

7 

30,8 

30,1 

29,4 

109 

743 

782 

822 

802 

902 

941 

981 

,021 

,060 

,100 

8 

35,2 

34,4 

33,6 

110 

04  139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

9 

39,6 

38,7 

37,8 

111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

41 

40 

39 

112 

922 

961 

999 

,038 

,077 

,115 

,154 

,192 

,231 

,269 

1 

4,1 

4,0 

3,9 

113 

05  308 

346 

385 

423 

461 

500 

538 

576 

614 

■652 

2 

8,2 

8,0 

7,8 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

,032 

3 

12,3 

12,0 

11,7 

115 

06  070 

108 

145 

183 

221 

258 

296 

333 

371 

'408 

4 

16,4 

16,0 

15,6 

116 

446 

«3 

521 

558 

695 

633 

670 

707 

744 

781 

5 

20,5 

20,0 

19,5 

117 

819 

856 

893 

930 

967 

,004 

,041 

,078 

,115 

,151 

6 

24,6 

24,0 

23,4 

118 

07  ISS 

225 

262 

298 

335 

372 

408 

445 

482 

518 

7 

28,7 

28,0 

27,3 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

8 

32,8 

32,0 

31,2 

120 

918 

954 

990 

,027 

,063 

,099 

,135 

,171 

,207 

,243 

9 

36,9 

36,0 

35,1 

121 

08  279 

314 

350 

386 

422 

458 

493 

529 

565 

600 

88 

37 

36 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

1 

3,8 

3,7 

3,6 

123 

991 

,026 

,061 

,096 

,132 

,167 

,202 

,237 

,272 

,307 

2 

7,6 

TA 

7,2 

124 

09  342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

3 

11,4 

11,1 

10,8 

125 

691 

728 

760 

795 

830 

864 

899 

934 

968 

,003 

4 

15,2 

14,8 

14,4 

120 

10  037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

5 

19,0 

18,5 

18,0 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

6 

22,8 

22,2 

21,6 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

,025 

7 

26,6 

25,9 

25,2 

129 

11  059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

8 

30,4 

29,6 

28,8 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 

9 

34,2 

33,3 

32,4 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

,024 

35 

34 

33 

132 

12  057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

1 

3,5 

3,4 

3,3 

133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

2 

7,0 

6,8 

6,6 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

,001 

3 

10,5 

10,2 

9,9 

135 

13  033 

066 

■   098 

130 

162 

194 

226 

258 

290 

322" 

4 

14,0 

13,6 

13,2 

136 

354 

386 

418 

450 

.  481 

513 

545 

577 

609 

640 

5 

17,5 

17,0 

16,5 

137 

672 

704 

735 

707 

799 

830 

862 

893 

925 

956 

6 

21,0 

20,4 

19,8 

138 

988 

,019 

,051 

,082 

,114 

,145 

,176 

,208 

,239 

,270 

7 

24,5 

23,8 

23,1 

139 

14  301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

8 

28,0 

27,2 

26,4 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

9 

31,5 

30,6 

29,7 

141 

922 

953 

983 

,014 

,045 

,076 

,106 

,137 

,168 

,198 

33 

31 

30 

142 

15  229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

1 

3,2 

3,1 

3,0 

143 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

2 

6,4 

6,2 

6,0 

144 

836 

866 

897 

927 

957 

987 

,017 

,047 

,077 

,107 

3 

9,6 

9,3 

9'0 

145 

16  137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

4 

12,8 

12,4 

12,0 

146 

435 

465 

.  495 

524 

554 

584 

613 

643 

673 

702 

5 

16,0 

15,5 

15,0 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

6 

19,2 

18,6 

18,0 

148 

17  026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

7 

22,4 

21,7 

21,0 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

8 

25,6 

24,8 

24,0 

150 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

9 

28,8 

27,9 

27,0 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P. 

logaeithms  of  numbees. 


235 


Table  XXXV. — Containing  logarithms  of  numbers  from  1  to  11,000 — Continued. 
[Extracted  from  Graass'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

ISO 

17  609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

151 

898 

926 

955 

984 

,013 

,041 

,070 

,099 

,127 

.156 

29 

28 

152 

18  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

1 

2,9 

2,8 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

2 

5,8 

5,6 

154 

752 

780 

808 

837 

865 

893 

921 

949 

977 

,005 

3 

8,7 

8,4 

155 

19  033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

4 

11,6 

11,2 

156 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

5 

14,5 

14,0 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

6 

17,4 

16,8 

158 

866 

893 

921 

948 

976 

,003 

,030 

,058 

,085 

,112 

7 

20,3 

19,6 

159 

20  140 

167 

194 

232 

249 

276 

303 

330 

368 

385 

8 

23,2 

22,4 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

9 

26,1 

25,2 

161 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 

27 

2e 

162 

952 

978- 

..005 

,032 

,059 

,085 

,112 

.139 

,165 

,192 

1 

2,7 

2,6 

163 

21  219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

2 

5,4 

6,2 

164 

4S4 

511 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8,1 

7,8 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10,8 

10,4 

166 

22  on 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 

13,5 

13,0 

167 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

6 

16,2 

15,6 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

7 

18,9 

18,2 

169 

789 

814 

840 

866 

891 

917 

943 

968 

994 

,019 

8 

21,6 

20,8 

170 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

9 

24,3 

23,4 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

25     1 

172 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

1 

2,5 

173 

805 

830 

855 

880 

905 

930 

955 

980 

,005 

,030 

2 

5,0 

174 

24  055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

3 

7,5 

175 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

4 

10,0 

176 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 

5 

12,5 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993 

,018 

6 

15,0 

178 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

7 

17,5 

179 

285 

310 

331 

358 

382 

406 

431 

455 

479 

503 

8 

20,0 

ISO 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

9 

22,5 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

2S 

182 

26  007 

031 

055 

079 

102 

120 

150 

174 

198 

221 

1 

2,4 

2,3 

183 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

2 

4,8 

4,6 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7,2 

6,9 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

9,6 

9,2 

186 

951 

975 

998 

,021 

,045 

,068 

,091 

,114 

,138 

,161 

5 

12,0 

11,6 

187 

27  184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

6 

14,4 

13,8 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

7 

16,8 

16,1 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 

8 

19,2 

]8,4 

190 

875 

898 

921 

944 

967 

989 

,012 

,035 

*058 

,081 

9 

21,6 

20,7 

191 

28  103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

22 

21 

192 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

1 

2,2 

2,1 

193 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4,4 

4,2 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

3 

6,6 

6,3 

195 

29  003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

4 

8,8 

8,4 

196 

226 

248 

270 

.292 

314 

336 

358 

380 

403 

425 

5 

11,0 

10,5 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

6 

13,2 

12,6 

198 

667 

688 

710 

732 

754. 

776 

798 

820 

842 

863 

7 

15,4 

14,7 

199 

885 

907 

929 

951 

973 

994 

,016 

,038 

,060 

*081 

8 

17,6 

16,S 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

9 

.19,8 

18,9 

N. 

L.O 

1 

2 

3 

* 

5 

6 

7 

8 

^ 

P.P. 

236 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXV. — Containing  logariihms  of  numbers  from  1  to  11,000- 
[Estractetl  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

201 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

22 

21 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

1 

2,2 

2,1 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

2 

4,4 

4,2 

204 

963 

984 

,006 

,027 

,048 

,069 

,091 

,112 

,133 

,154 

3 

6,6 

6,3 

205 

31  175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

4 

8,8 

8,4 

206 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

5 

11,0 

10,5 

207 

597 

618 

639 

660 

681 

703 

723 

744 

765 

785 

6 

13,2 

12,6 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

7 

15,4 

14,7 

209 

32  015 

035 

.  056 

077 

098 

118 

139 

160 

181 

201 

8 

17,6 

16,8 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

9 

19,8 

18,9 

2U 

428 

449 

469 

490 

610 

531 

552 

572 

593 

613 

20    1 

212 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

1 

2,0 

213 

838 

858 

879 

899 

919 

940 

960 

980 

,001 

,021 

2 

4,0 

214 

33  041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

3 

6,0 

215 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

4 

8,0 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

5 

10,0 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

6 

12,0 

218 

846 

866 

885 

905 

925 

945 

965 

985 

,005 

,025 

7 

14,0 

219 

3i  044 

061 

084 

104 

124 

143 

163 

183 

203 

223 

8 

16,0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

9 

18,0 

221 

439 

459 

479 

498 

518 

%m 

557 

577 

596 

616 

19    1 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

1 

1,9 

323 

830 

850 

869 

889 

908 

928 

947 

967 

986 

,005 

2 

3,8 

224 

35  025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

3 

5,7 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

4 

7,6 

226 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

5 

9,5 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

6 

11,4 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

7 

13,3 

"29 

984 

,003 

,021 

,040 

,059 

,078 

,097 

,116 

,135 

,154 

8 

15,2 

230 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

'342 

9 

17,1 

231 

361 

380 

399 

418 

436 

455 

474 

493 

511 

530 

18 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

1 

1,8 

233 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

2 

3,6 

234 

922 

940 

959 

977 

996 

,014 

,033 

,051 

,070 

,088 

3 

5,4 

235 

37  107 

125 

144 

162 

181 

199 

218 

236 

254 

'273 

,  4 

7,2 

236 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

5 

9,0 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

6 

10,8 

238 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

7 

12,6 

239 

840 

858 

876 

894 

912 

931 

949 

967 

985 

003 

8 

14,4 

240 

38  021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

9 

16,2 

2a 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

17    1 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

1 

1,7 

243 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

2 

3,4 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

890 

3 

5,1 

245 

917 

034 

952 

970 

987 

,005 

,023 

,041 

,058 

,076 

4 

?»* 

246 

39  094 

111 

129 

146 

164 

182 

199 

217 

235 

252 

5 

8,5 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

6 

10,2 

248 

445 

463 

480 

498 

515 

533 

550 

568 

685 

602 

7 

11,9 

249 

620 

637 

655 

072 

690 

707 

724 

742 

759 

777 

8 

13,6 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

9 

15,3  ■ 

K. 

L.  0. 

1 

2 

3 

4 

5 

6 

■  7 

8 

9 

P.P. 

LOGAEITHMS  OF  NUMBERS. 


237 


Table  XXXV. — Containimj  logarithms  of  mimhers  from  I  to  11,000 — Continued. 
[Extracted  from  Gauss'  Loo;arithmic  ami  Trigonometric  Tablef&.J 


N. 

L.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

250 

39  794 

811 

829 

846 

863' 

881 

898 

915 

933 

950 

251 

967 

985 

,002 

,019 

,037 

,054 

.071 

,088 

,106 

,123 

1 

1,8 

252 

40140 

157 

175 

192 

209 

220 

243 

261 

278 

295 

2 

3,6 

253 

312 

329 

346 

364 

381 

398 

415 

432 

449- 

466 

3 

6,4 

254 

483 

500 

518 

535 

552 

509 

586 

603 

620 

637 

4 

7,2 

265 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

5 

9,0 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

•976 

6 

10,8 

257 

993 

,010 

.027 

,044 

,061 

,078 

,095 

,111 

,128 

,145 

7 

12,6 

258 

41 162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

8 

14,4 

259 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

9 

16,2 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 

„     1 

261 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

1 

1/7 

262 

83"0 

847 

863 

880 

896 

913 

929 

946 

963 

979 

2 

3,4 

263 

996 

,012 

,029 

.,045 

,062 

,078 

,095 

.,111 

,127 

,144 

3 

5,1 

264 

42160 

177 

193 

210 

220 

243 

259 

275 

292 

308 

4 

6,8 

265 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

5 

8,5 

266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

6 

10,2 

267 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

7 

11,9 

268 

813 

830 

846 

862 

.   878 

894 

911 

927 

943 

959 

8 

13,6 

269 

975 

991 

,008 

,024 

,040 

,056 

,072 

,088 

,104 

,120 

9 

15,3 

270 

43136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

,/ 

271 

297 

313 

329 

345 

361 

.'  377 

393 

409 

425 

441 

1 

1,6 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

■  600 

2 

3,2 

273 

616 

632 

048 

664 

680 

696 

712 

727 

743 

759 

2 

4,8 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

4 

6,4 

275 

933 

949 

965 

981 

996 

,012 

,028 

.,044 

,059 

,075 

5 

8,0 

276 

44  091 

107 

122 

138 

154 

170 

186 

'201 

217 

232 

6 

9,6 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

7 

11,2 

278 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

8 

12,8 

279 

560 

576 

592 

607 

623 

638 

654 

669 

686 

■   700 

9 

14,4 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

,.     1 

281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

,010 

1 

1,5 

i82 

45  025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

2 

3,0 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

3 

4,5 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

4 

6,0 

285 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

5 

7,5 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

0 

9,0 

287 

788 

803 

818 

834 

849 

864 

879 

894 

009 

924 

7 

10,5 

288 

939 

954 

969 

984 

.,000 

,016 

,030 

,045 

,060 

.,075 

8 

12,0 

289 

46  090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

9 

13,5 

290 

240 

255 

270 

286 

300 

315 

330 

345 

359 

374 

,,     1 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

1 

1,4 

292 

538 

553 

668 

583 

598 

613 

627 

642 

657 

672 

2 

2,8 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

3 

4,2 

294 

■835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

4 

5,6 

295 

982 

997 

,012 

.,026 

..041 

,056 

,070 

,085 

,100 

,114 

5 

7,0 

296 

47129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

6 

8,4 

297 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

7 

9,8 

298 

422 

436 

451 

465 

480 

494 

509 

524 

538 

553 

8 

11,2 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

9 

12,6 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

M". 

.L.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

238 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XX.XV. — Containing  logarithms  of  iiamiers  from  1  to  11,000 — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 
300 

L.   0 

47  712 

1 

2 

3  ■ 

4 

= 

6 

7 

8 

9 

P.P. 

727 

741 

756 

770 

784 

799 

813 

828 

842 

301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

302 

48  001 

015 

029 

044 

058 

073 

087 

101 

116 

130 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

15 

30 1 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 

1 

1,5 

305 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

o 

3,0 

306 

572 

■  586 

601 

615 

629 

643 

657 

671 

686 

700 

3 

4,5 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

4 

6,0 

308 

855 

869 

883 

897 

911 

926 

940 

954 

963 

982 

5 

7,5 

309 

996 

,010 

,024 

,038 

,052 

,066 

,080 

,094 

,108 

,122 

6 

9,0 

310 

49  136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

7 
8 

10,5 
12,0 

311 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

9 

13,5 

312 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 

313 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 

314 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

315 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

11 

316 

969 

982 

996 

,010 

,024 

,037 

,051 

,065 

,079 

,092 

1 

1,4 

317 

50  106 

120 

133 

147 

101 

174 

188 

202. 

215 

229 

2 

2,8 

318 

243 

256 

270 

284 

297 

311 

325 

338 

352 

365 

3 

4,2 

319 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

4 

5,6 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

5 
6 

7,0 
8,4 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

7 

9,8 

322 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

8 

11,2 

323 

920 

934 

947 

961 

974 

987 

,001 

,014 

,028 

,041 

9 

12,6 

324 

51  055 

068 

081 

095 

108 

121 

135 

148 

162 

175 

325 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 

326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

13 

327 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 

1 

1,3 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

2 

2,6 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

3 

3,9 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

4 
5 

5,2 
6,5 

331 

983 

996 

,009 

jm 

,035 

,048 

4)61 

,075 

,088 

,101 

6 

7,8 

332 

52  114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

7 

9,1 

333 

244 

257 

270 

284 

297 

310 

•  323 

336 

349 

362 

8 

10,4 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

9 

11,7 

335 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 

1 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 

12      1 

337 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 

1 

1,2 

338 

892 

905 

917 

930 

943 

956 

969 

982 

994 

,007 

2 

2,4 

339 

53  020 

033 

046 

058 

071 

084 

097 

110 

122 

135 

3 

3,6 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

4 
5 

4,8 
6,0 

341 

275 

288 

301 

314 

326 

339 

352 

364 

377 

390 

6 

7,2 

342 

403 

415 

428 

441 

453 

466 

479 

491 

504 

517 

7 

8,4 

343 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

8 

9,6 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

.  9 

10,8 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

346 

908 

920 

933 

945 

958 

970 

983 

995 

,008 

,020 

347 

54  033 

045 

058 

070 

083 

095 

108 

120 

133 

145 

348 

158 

170 

183 

195 

208 

220 

233 

245 

238 

270 

349 

283 

295 

307 

820 

332 

345 

357 

370 

382 

394 

350 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

N. 

L.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

LOGARITHMS  OF  NUMBEES. 


239 


Table  XXXV". — Containing  logarithms  of  numbers  from  1  to  11,000 — Coutiuued. 
[Extracted  from  G-ausa'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

350 

51  407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

351 

531 

643 

656 

568 

580 

693 

605 

617 

630 

642 

352 

654 

667 

679 

691 

704 

716 

728 

741 

753 

765 

13 

353 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 

354 

900 

913 

926 

937 

949 

962 

974 

986 

998 

,011 

1 

1,3 

355 

55  023 

035 

047 

060 

072 

084 

096 

108 

121 

133 

2 

2,6 

356 

145 

167 

169 

182 

194 

206 

218 

230 

2J2 

255 

3 

3,9 

357 

267 

279 

291 

303 

315 

328 

340 

362 

364 

376 

4 

5,2 

358 

388 

40O 

413 

■425 

437 

449 

461 

473 

486 

497 

5 

6,5 

359. 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

6 

7,8 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

7 
8 

9,1 
10,4 

361 

751 

763 

775 

787 

799 

811 

823 

836 

847 

859 

9 

11,7 

362 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 

363 

991 

,003 

,016 

,027 

,038 

.050 

,062 

,074 

,086 

,098 

364 

56  110 

122 

134 

146 

158 

170 

182 

194 

205 

217 

12 

305 

229 

241 

253 

265 

277 

289 

301 

312 

321 

336 

366 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

1 

1,2 

367 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

2 

2,4 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

3 

3,6 

369 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

4 

4,8 

370 

820 

832 

844 

856 

867 

879 

891 

902 

914 

926 

5 
6 

6,0 
7,2 

371 

937 

949 

961 

972 

984 

996 

,008 

,019 

,031 

,043 

7 

8,4 

372 

67  054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

8 

9,6 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

9 

10,8 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

375 

403 

415 

426 

438 

449 

461 

473 

434 

496 

507 

376 

619 

630 

642 

553 

665 

576 

688 

600 

611 

623 

11 

377 

634 

646 

657 

669 

630 

692 

703 

715 

726 

738 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

1 

1,1 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

2 

2,2 

380 

978 

990 

,001 

,013 

,024 

,035 

,047 

,058 

,070 

,081 

3 
4 

3,3 

4,4 

381 

58  092 

104 

116 

127 

138 

149 

161 

172 

184 

196 

5 

6,5 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

6 

6,6 

383 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

7 

7,7 

384 

433 

444 

456 

467 

478 

490 

501 

512 

624 

535 

8 

8,8 

385 

646 

657 

569 

580 

591 

602 

614 

625 

636 

647 

9 

9,9 

386 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 

387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 

388 

883 

894 

906 

917 

928 

939 

960 

961 

973 

984 

10 

389 

995 

»ao6 

,017 

,028 

,040 

,051 

,062 

,073 

,084 

,095 

390 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

1 

2 

1,0 
2,0 

391 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318 

3 

3,0 

392 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

4 

4,0 

393 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

5 

5,0 

394 

550 

661 

572 

683 

694 

6C5 

616 

627 

638 

649 

6 

6,0 

395 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

7 

7,0 

396 

770 

780 

701 

802 

813 

824 

835 

846 

857 

868 

8 

8,0 

397 

879 

800 

901 

912 

923 

934 

946 

956 

966 

977 

9 

9,0 

398 

988 

999 

,010 

,021 

,032 

,043 

,054 

,065 

,076 

,086 

399 

60  097 

108 

119 

130 

141 

152 

163 

173 

184 

195 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

240 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXXY.— Containing  logarithms  of  mimbers  from  1  to  i^OW— Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


3,0 
4,0 
5,0 
6,0 
7,0 
8,0 
9,0 


LOGAEITHMS  OF  NUMBEES. 


241 


Table  XXXV. — Containing  logarithms  of  numhera  from  1  to  11,000 — Continued. 
[Extracted  from  Gauss'  Logaritlimic  and  Triganometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

450 

65  321. 

331 

341 

350 

360 

369 

379 

389 

398 

408 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 

452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

454 

70G 

715 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

456 

896 

906 

916 

925 

935 

944 

954 

903 

973 

982 

10 

457 

992 

,001 

*011 

,020 

,030 

,039 

,049 

,058 

,068 

,077 

1 

1,0 

458 

66  087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

2 

2,0 

459 

181 

191 

200 

.  210 

219 

229 

238 

247 

257 

266 

3 

3,0 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

4 
5 

4,0 
5,0 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

6 

6,0 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

7 

7,0 

463 

558 

507 

577 

586 

596 

605 

614 

624 

633 

642 

8 

8,0 

464 

■   652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

9 

9,0 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

467 

932 

941 

950 

960 

969 

978 

987 

997 

,006 

,015 

468 

67  025 

014 

043 

052 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

•   25G 

265 

274 

284 

293 

„ 

471 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 

1 

0,9 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

2 

1,8 

473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

3 

2,7 

474 

578 

687 

596 

605 

614 

624 

633 

642 

651 

660 

4 

3,6 

475 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

5 

4,5 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

6 

5,4 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

7 

6,3 

478 

943 

952 

961 

970 

979 

988 

997 

,006 

,015 

,024 

8 

7,2 

479 

68  034 

043 

052 

061 

070 

079 

088 

097 

106 

115 

9 

8,1 

480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

305 

314 

323 

332 

341 

350 

3')9 

368 

377 

386 

483 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

485 

574 

583 

592- 

601 

010 

619 

628 

637 

646 

655 

S 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

1 

0,8 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

2 

1,0 

488 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

3 

2,4 

489 

931 

940 

949 

958 

966 

975 

984 

993 

,002 

,011 

4 

3,2 

490 

69  020 

028 

037 

046 

055 

064 

078 

082 

090 

099 

5 
6 

4,0 
4,8 
5,6 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

7 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

8 

6,4 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

9 

7,2 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

062 

671 

679 

°688 

697 

705 

714 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

MON  XXII- 


-16 


242 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXXV. — Containiiifj  logarithms  of  numbers  from  1  to  11,000 — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

600 

69,  897 

906 

914 

923 

932 

940 

949 

958 

906 

975 

501 

984 

992 

,001 

,010 

,018 

,027 

,036 

,044 

,053 

,062 

502 

70,  070 

079 

088 

096 

105 

114 

122 

131 

140 

148 

503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 

504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 

505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 

9 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

1 

0,9 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

2 

1,8 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

3 
4 

2,7 
3,6 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

5 

4,5 

512 

927 

935 

944 

952 

901 

969 

978 

980 

995 

,003 

6 

5,4 

513 

71,012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

7 

6,3 

514 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

8 

7,2 

51-5 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 

9 

8,1 

516 

265 

273 

282 

29U 

299 

307 

315 

324 

332 

341 

517 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 

518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 

519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 

620 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 

8 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

1 

0,8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

926 

2 

1,6 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

,008 

3 

2,4 

525 

72,  016 

024 

032 

041 

049 

957 

066 

074 

082 

090 

4 

3,2 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5 

4,0 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

6 

4,8 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

7 

5,6 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

8 

6,4 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

601 

9 

7,'. 

531 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 

532 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 

533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

534 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 

535 

835 

843 

852 

860 

869 

876 

884 

892 

900 

908 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

J 

537 

997 

,006 

,014 

,022 

,030 

,038 

,046 

,054 

,062 

,070 

1 

0,7 

538 

73,  078 

■   086 

094 

102 

111 

119 

127 

135 

143 

151 

2 

1,4 

639 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

3 

2,1 

640 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

4 
5 

2,8 
3,5 

541 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

6 

4,2 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

7 

4,9 

543 

480 

488 

496 

504 

512 

520 

528 

536 

644 

552 

8 

5,6 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

9 

6,3 

545 

640 

648 

656 

664 

673 

679 

687 

695 

703 

711 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

547 

799 

807 

815 

823 

■630 

838 

846 

854 

862 

870 

548 

87S 

886 

894 

902 

910 

918 

926 

933 

941 

949 

549 

957 

965 

973 

981 

989 

997 

*005 

*013 

*020 

*028 

650 

74,  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

K. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

,  8 

9 

P.P. 

LOGARITHMS  OF  IvTUMBBRS. 


243 


Table  XXXY. — Containing  logarithms  of  numbers  from  1  to  ll,00t 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables." 


N. 

L.  0. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

551 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 

552 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 

553 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 

554 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 

555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 

556 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

553 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

559 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

561 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 

R 

562 

974 

981 

989 

997 

,005 

^012 

,020 

.028 

,035 

.043 

1 

0,8 

563 

75  051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

2 

1,6 

564 

128 

136 

143 

151 

159 

166 

174 

■   182 

189 

197 

3 

2,4 
3,2 

565 

205 

213 

220 

228 

230 

243 

261 

259 

266 

274 

4 

566 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

5 

4,0 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

6 

4,8 

568 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

7 

5,6 

569 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 

8 

6,4 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

9 

7,2 

571 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

572 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 

563 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

575 

967 

974 

982 

989 

997 

,005 

,012 

.020 

.027 

,035 

576 

76  042 

050 

057 

065 

072 

080 

087 

095 

103 

110 

577 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 

578 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 

579 

268 

275 

283 

290 

298 

305 

313 

320 

328 

335 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

7 
1  0,7 

581 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 

582 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

2 

1,4 

683 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

3 

2,1 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

4 

2,8 

585 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

5 

3,5 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

6 

4,2 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

7 

4,9 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997 

,004 

8 

5,6 

589 

77  012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

9 

6,3 

690 

086 

093 

100 

107 

115 

122 

129 

137 

144 

151 

591 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

593 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 

594 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 

595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

596 

525 

532 

539 

646 

554 

561 

568 

576 

583 

590 

597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

599 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 

600 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

S. 

L.  0. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

244 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXV. — Containing  logarithms  of  numbers  from  1  to  11,000 — Coutinned. 
[Extracted  from  G.iuss'  Log.arithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

601 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 

602 

960 

967 

974 

981 

988 

^96 

,003 

,010 

,017 

,025 

603 

78  032 

039 

046 

053 

061 

068 

075 

082 

089 

097 

604 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 

605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

8 

607 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

608 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

1 

0,8 

6oa 

462 

469 

470 

483 

490 

497 

504 

512 

519 

526 

2 

1,6 

610 

533 

540 

547 

554 

561 

669 

576 

583 

590 

597 

3 
4 

2,4 
3,2 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

6 

4,0 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

6 

4,8 

era 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

7 

5,6 

614 

817 

824 

831 

888 

845 

852 

859 

866 

873 

880 

8 

6,4 

615 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 

9 

7,2 

616 

958 

965 

072 

979 

986 

993 

,000 

,007 

,014 

,021 

617 

79  029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

618 

099 

106 

113 

120 

127 

134 

141 

148 

155 

163 

6in 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

621 

309 

316 

323 

330 

337 

344 

351 

358 

360 

372 

7 

822 

379 

386 

393 

4U0 

407 

414 

421 

428 

435 

442 

1 

0,7 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

2 

l,* 

624 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

3 

2,1 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

4 

2,8 

626 

657 

664 

.  671 

678 

685 

692 

699 

706 

713 

720 

5 

3,5 

627 

727 

734 

741 

748 

754 

761 

768 

775 

782 

780 

6 

4,2 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

7 

4,9 

629 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 

8 

6'6 

030 

934 

941 

948 

955 

962 

969 

976 

982 

989 

996 

9 

6,3 

631 

80  003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

632 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 

633 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

6 

635 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

1 

0,6 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

2 

1,2 

638 

482 

489 

496 

502 

509 

516 

623 

530 

536 

543 

3 

1,8 

639 

550 

557 

564 

570 

577 

584 

591 

698 

604 

611 

4 

2,4 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

5 
6 

3,0 
3,6 

641 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

7 

4,2 

642 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

8 

4,8 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

9 

5,4 

644 

889 

895 

902 

909 

916 

■922 

929 

936 

943 

949 

645 

956 

903 

069 

976 

983 

990 

996 

,003 

,010 

,017 

646 

81  023 

030 

037 

043 

050 

057 

064 

070 

077 

084 

647 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 

648 

138 

164 

171 

178 

184 

191 

198 

204 

211 

218 

649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

650 

291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

N. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGARITHMS  OF  NUMBEES. 


245 


Table  XXXY.— Containing  logarithms  of  niimbcrx  from  1  to  11,000— Continued. 
[Extracted  from  Gauss'  Logaritlimic  and  Trigouometric  Tables.] 


8 

9 

315 

351 

411 

418 

4V8 

485 

bU 

551 

611 

617 

till 

681 

743 

750 

809 

816 

8Vb 

883 

941 

948 

,007 

,014 

073 

079 

138 

145 

204 

210 

269 

276 

334 
400 

341 
inR 

246 


A  MANUAL  OF  TOPOGEAPHIC.METHODS. 


Table  XXXV. — Containing  logarithms  of  numbers  from  1  to  11,000 — Contmued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5   . 

6 

7 

8 

9 

P.  P. 

•sm 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

701 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

702 

634 

640 

646 

052 

658 

665 

671 

677 

683 

689 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 

704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

f  ' 

705 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

1 

706 

■  880 

887 

893 

899 

905 

911 

917 

924 

930 

936 

1 

0,7 

707 

942 

948 

934 

960 

967 

973 

979 

985 

991 

997 

2 

1,4 

708 

85  003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

3 

2,1 

709 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

4 

2,8 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

5 
6 

3,5 
4,2 

711 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

7 

4,9 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

8 

5,6 

713 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

9 

6,3 

714 

370 

376 

382  . 

388 

394 

400 

406 

412 

418 

425 

715- 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

716 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

718 

612 

618 

625 

631 

637 

643 

649 

655 

601 

667 

719 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

e 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

1 

0,6 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

2 

1,2 

723 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

3 

1,8 

724 

974 

980 

986 

992 

998 

,004 

,010 

,.016 

,022 

,028 

4 

2,4 

725 

86  034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

5 

3,0 

726 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

6 

3,6 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

7 

4,2 

728 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

8 

4,8 

729 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

9 

5,4 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

731 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

732 

451 

457 

463 

469 

475 

481 

487 

493 

499 

604 

733 

510 

516 

522 

528 

534 

540 

546 

552 

658 

664 

734 

570 

576 

581 

587 

593 

599 

605 

611 

617 

622 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

736 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

5 

737 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

738 

806 

812 

817 

82a 

829 

835 

841 

847 

853 

859 

1 

0,5 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

2 

1,0 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

3 
4 

1,5 
2,0 

741 

982 

988 

994 

999 

,,005 

,011 

»017 

,023 

,029 

,035 

5 

2,5 

742 

87  040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

6 

3,0 

743 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

7 

3,5 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

8 

4,0 

745 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 

9 

4,5 

746 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

748 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 

749 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

If. 

L.   0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGAEITHMS  OF  NUMBEES. 


247 


Table  XXXV. — Containing  logarithms  of  nunibers  from  1  to  llfiOO — Continued. 

[Extracted  from  Gauss'  Logaritlimic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

750 

87  506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

751 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 

752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 

753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

^ 

756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

757 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 

758 

967 

973 

978 

984 

990 

996 

,001 

,007 

,013 

,018 

759 

88  024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 

761 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 

6 

762 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

1 

0,6 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

2 

1,2 

764 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

3 

1,8 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

4 

2,4 

766 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

5 

3,0 

767 

480 

485 

491 

497 

502 

508 

513 

619 

525 

530 

6 

3,6 

768 

536 

542 

547 

553 

659 

564 

570 

676 

581 

537 

7 

4,2 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

8 

4,8 

770 

«e49 

655 

660 

666 

672 

677 

■  683 

689 

694 

700 

9 

6'4 

771 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 

772 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

775 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997 

,003 

,009 

,014 

,020 

,025 

,031 

,037 

777 

89  042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

778 

098 

104 

109 

115 

120 

126 

131 

137 

143 

143 

779 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

781 

265 

271 

276 

282 

287' 

293 

298 

304 

310 

315 

5 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

1 

0,5 

783 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

2 

I'O 

784 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

3 

1,5 

785 

487 

492 

498 

504 

509 

515 

520 

626 

531 

537 

4 

2,0 

786 

542 

648 

553 

559 

564 

570 

575 

.   581 

586 

592 

5 

2,5 

787 

597 

603 

609 

614 

620 

625 

631 

638 

642 

647 

6 

3,0 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

7 

3,5 

789 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 

8 

4,0 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

9 

4,5 

791 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 

792 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

794 

982 

988 

993 

998 

,004 

,009 

,015 

,020 

«026 

,031 

795 

90  037 

042 

048 

053 

059 

064 

069 

075 

080 

086 

796 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 

797 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

248 


A  MANUAL  OF   TOPOGEAPHIC  METHODS. 


Table  XXXV. — Containing  logarithms  of  numiers  from  1  to  il,000— Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0   1   1 

'■•,:'. 

4 

_ 

6 

' 

8 

9 

r.  P. 

800 

90  309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

801 

"  363 

369 

374 

380 

385 

390 

396 

401 

407 

412 

802 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 

803 

472 

477 

482 

488 

493 

499 

504 

509 

515 

620 

804 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 

805 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

*   687 

693 

698 

703 

709 

714 

720 

725 

730 

736 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 

809 

795 

800 

806 

811 

816 

822 

827 

832 

838 

843 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

811 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 

6 

813 

956 

961 

966 

972 

977 

982 

988 

993 

998 

,004 

1 

0,6 

813 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

2 

1,2 

814. 

062 

068 

073.,- 

078 

084 

089 

094 

100 

105 

110 

3 

1,8 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

4 

2,4 

816 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

5 

3,0 

817 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

6 

3,6 

818 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

7 

4,2 

819 

328 

334 

339 

344 

350 

355- 

360 

365 

371 

376 

8 

4,8 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

9 

5,4 

821 

434 

440 

445 

450 

455 

461 

466 

471 

477 

482 

» 

822 

487 

492 

498 

503 

508 

514 

519 

624 

529 

535 

823 

540 

545 

551 

556 

561 

S66 

572 

577 

582 

587 

824 

593 

598 

603 

609 

614 

619 

624 

030 

635 

640 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 

827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

934 

840 

845 

850 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

sso 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

831 

960 

965 

971 

976 

981 

986 

991 

997 

,002 

,007 

6 

832 

92  012 

018 

023 

028 

033 

038 

044 

049 

054 

059 

1 

0,5 

833 

065 

070 

075 

080 

085 

091 

096 

101 

106 

111 

2 

1,0 

834 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

3 

1,5 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

4 

2,0 

83<> 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

5 

2,5 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

6 

3,0 

838 

324 

330 

835 

340 

345 

350 

355 

361 

366 

371 

7 

3,5 

839 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

8 

4,0 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

9 

4,5 

841 

480 

485 

490 

495 

500 

505 

511 

516 

521 

526 

842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 

843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 

844 

634 

639 

145 

650 

655 

660 

665 

670 

675 

681 

845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

782 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

850 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

IJ". 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P.. 

LOGAEITHMS  OF  NUMBEES. 


249 


Table  XXXV. — Containing  logarithms  of  numbers  from  1  to  lljOOO — Continued. 
[Extracted  from  Gauss'  Logarithmic  and.  Trigonometric  Tables.] 


N. 

L.  0 

1 

3 

4 

5 

6 

'  !  ' 

9 

I 

.P. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

851 

993 

998 

,003 

,008 

,013 

.018 

,024 

,029 

,034 

,039 

852 

93  044 

049 

054 

059 

064 

069 

075 

080 

085 

090 

853 

095 

100 

105 

110 

115 

120 

125 

131 

136 

141 

854 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 

855 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

857 

298 

303 

308 

313 

318 

323 

328 

334 

339 

344 

6 

858 

349 

.  354 

359 

364 

369 

374 

379 

384 

389 

394 

1 

0,6 

859 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

2 

1,2 

8G0 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

3 

4 

1,8 

2,4 

861 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

5 

3,0 

862 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

6 

3,6 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

7 

4,2 

864 

651 

656 

661 

666 

671 

676 

682 

687 

092 

697 

8 

4,8 

865 

702 

707 

713 

717 

722 

727 

732 

737 

742 

747 

9 

5,4 

866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

869- 

902 

907 

912 

917 

922  . 

927 

932 

937 

942 

947 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

871 

94  002 

007 

012 

017 

022 

027 

032 

037 

042 

047 

5 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

1 

0,5 

873 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

2 

1,0 

874 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

3 

1,5 

875 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

4 

2,0 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

5 

2,5 

877 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

6 

3,0 

878 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

7 

3,5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

8 

4,0 

SSO 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

9 

4,5 

881 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

882 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

1 

0,4 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

2 

0,8 

889 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

3 

1,2 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

4 
5 

1,6 
2,0 

891 

988 

993 

998 

,002 

,007 

,012 

,017 

,022 

,027 

,032 

6 

2,4 

892 

95  036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

7 

2,8 

893 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

8 

3,2 

894 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

9 

3,6 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 

898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

899 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

!«-. 

L.  0 

1 

2 

3 

4 

3 

0 

7 

8 

9 

P 

.P. 

250 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXV. — Containing  logaritlims  of  numbers  from  1  to  11,000. — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

900 

95  424 

429 

434 

439 

- 

444 

448 

453 

458 

463 

468 

901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

616 

902 

521 

525 

530 

635 

640 

545 

550 

554 

559 

564 

903 

669 

574 

578 

583 

588 

693 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

905 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

91p8 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

6 

912 

999 

»004 

,009 

«014 

,019 

,023 

,028 

,033 

,038 

,042 

1 

0,5     1 

913 

96  047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

2 

1,0     i 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

3 

1,5     1 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

4 

2,0 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

5 

2,5 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

6 

3,0 

918 

284 

289 

294 

298 

303 

308 

313 

817 

322 

327 

7 

3,5 

919 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

8 

4,« 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

9 

4,5 

921 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

501 

506 

611 

615 

923 

520 

525 

530 

534 

539 

544 

648 

553 

558 

562 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

662 

656 

926 

661 

006 

670 

675 

630 

685 

689 

694 

699 

703 

927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755 

759 

704 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

330 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

931 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

4 

932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

1 

0,4 

933 

988 

993 

997 

,002 

,007 

.011 

,016 

,021 

,025 

,030 

2 

0,8 

934 

97  035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

3 

1,2 

935 

081 

0S6 

090 

095 

100 

104 

109 

114 

118 

123 

4 

1,6 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

2,0 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

6 

2,4 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

7 

2,8 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

8 

3,2 

910 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

9 

3,6 

941 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 

942 

405 

410 

414 

419 

424 

42,S 

433 

437 

442 

447 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

945 

543 

548 

552 

667 

662 

566 

571 

675 

580 

585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

663 

658 

663 

667 

672 

676 

948 

681 

685 

690 

693 

699 

704 

708 

713 

717 

722 

949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

X.   0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

P.P. 

LOGARITHMS  OF  i^UMBEES. 


251 


Table  XXXV. — Containing  logarithms  of  mmibers  from  1  to  11,000,- 
[Estracted  fi'oiu  Gauss'  Logarithmic  aud  Trigonoroetric  Tables.] 


0 

1 

2 

3 

7  772 

777 

782 

786 

818 

823 

827 

832 

864 

868 

873 

877 

909 

914 

918 

923 

955 

959 

964 

968 

8  000 

005 

009 

014 

046 

050 

055 

059 

091 

096 

100 

105 

137 

141 

146 

150 

182 

186 

191 

195 

227 

232 

236 

241 

272 

277 

281 

286 

318 

322 

327 

331 

363 

367 

372 

376 

408 

412 

417 

421 

453 

457 

462 

466 

498 

502 

507 

511 

543 

547 

552 

556 

588 

692 

597 

601 

632 

637 

641 

640 

677 

682 

686 

691 

722 

726 

731 

735 

767 

771 

776 

780 

811 

816 

820 

825 

856 

860 

865 

869 

900 

905 

909 

914 

945 

949 

954 

958 

989 

994 

998 

,003 

99  034 

038 

043 

047 

078 

083 

087 

092 

123 

127 

131 

136 

167 

171 

176 

180 

211 

216 

220 

224 

255 

260 

264 

269 

300 

304 

308 

313 

844 

348 

352 

357 

388 

392 

396 

401 

432 

436 

441 

415 

252 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXV. — Containinfi  Ingarithms  of  numhers  from  1  to  ii,00fl.— Coutinuecl. 
[Extracted  from  Gauss'  Logarithmic  and  Trigouometric  Ta"bles-] 


N. 

L.   0 

1 

2 

3 

4 

5 

6 

7 


8 

9 

d. 

louo 

000  ouoo 

0434 

0869 

1303 

1737 

2171 

2605 

3039 

3472 

3907 

434 

1001 

4341 

4775 

5208 

5642  1 

6076 

6510 

6943 

7377 

7810 

8244 

434 

1002 

8677 

9111 

9544 

9977 

.0411 

,0844 

,1277 

,1710 

,2143 

,2576 

433 

1003 

001  3009 

3442 

3875 

4308 

4741 

5174 

5607 

6039 

0472 

6905 

433 

1004 

7337 

7770 

8202 

8C35 

9067 

9499 

9932 

,0364 

,0796 

.1228 

432 

1(106 

002  1601 

2093 

2525 

2957 

3389 

3821 

4253 

4685 

5116 

5548 

432 

1006 

5980 

6411 

6843 

7275 

7706 

8138 

8569 

9001 

9432 

9863 

431 

1007 

003  0295 

0726 

1157 

1588 

2019 

2451 

2882 

3313 

3744 

4174 

431 

lOOS 

4605 

5U36 

5467 

5898 

6328 

6759 

7190 

7620 

8051 

8481 

431 

10U9 

8912 

9342 

9773 

,0203 

,0633 

,1063 

,1493 

,1924 

,2354 

,2784 

430 

1010 

004  3214 

3644 

4074 

4504 

4933 

'5363 

5793 

6223 

6652 

7082 

430 

1011 

7512 

7941 

8371 

8800 

9229 

9659 

,0088 

,0517 

,0947 

,1376 

429 

1012 

005  1805 

2234 

2663 

3092 

3521 

3950 

4379 

4808 

5237 

5666 

429 

1013 

6094 

6523 

6952 

7380 

7809 

8238 

8666 

9094 

9523 

9951 

429 

1014 

006  03S0 

0808 

1236 

1664 

2092 

2521 

2949 

3377 

3805 

4233 

428 

1015 

4660 

5088 

5516 

5944 

6372 

6799 

7227 

7655 

8082 

8510 

428 

1010 

8937 

9365 

9792 

,0219 

,0647 

,1074 

,1501 

,1928 

,2355 

,2782 

427 

1017 

007  3210 

3637 

4064 

4490 

4917 

5344 

5771 

6198 

6624 

7051 

427 

1018 

7478 

7904 

8331 

8757 

9184 

9610 

,0037 

,0463 

,0889 

,1316 

426 

1019 

008  1742 

2168 

2594 

3020 

3446 

3872 

4298 

4724 

'5150 

5576 

426 

1020 

6002 

6427 

6853 

7279 

7704 

8130 

8556 

8981 

9407 

9832 

426 

1021 

009  0257 

0683 

1108 

1533 

1959 

2384 

2809 

3234 

3659 

4084 

425 

1022 

4509 

4934 

5359 

5784 

6208 

6633 

7058 

7483 

7907 

8332 

425 

102:1 

8756 

9181 

9605 

.0030 

.0454 

,0878 

,1303 

,1727 

,2151 

,2575 

424 

1024 

010  3000 

3424 

3848 

4272 

4696 

5120 

5544 

5967 

6391 

6815 

424 

1025 

7239 

7662 

8086 

8510 

8933 

9357 

9780 

,0204 

,0627 

,1050 

424 

1026 

Oil  1474 

1897 

2320 

2743 

3166 

3590 

4013 

4436 

4859 

5282 

423 

1027 

5704 

6127 

6550 

6973 

7396 

7818 

8241 

8664 

9086 

9509 

423 

1028 

9931 

.0854 

.,0776 

,1198 

,1621 

,2043 

,2465 

.2887 

,3310 

,3732 

422 

1029 

012  4154 

4576 

4998 

5420 

'5842 

6264 

6685 

7107 

7529 

7951 

422 

1030 

8372 

8794 

9215 

9637 

,0059 

,0480 

,0901 

,1323 

,1744 

,2165 

422 

1031 

013  2587 

3008 

3429 

3850 

4271 

4692 

5113 

5534 

5955 

6376 

421 

1032 

6797 

7218 

7639 

8059 

8480 

8901 

9321 

9742 

,0162 

,0583 

421 

1033 

014  1003 

1424 

1844 

2264 

2685 

3105 

3525 

3945 

4365 

4785 

420 

1034 

5205 

5625 

6045 

6465 

6885 

7305 

7725 

8144 

8564 

8984 

420 

1035 

9403 

9823 

,0243 

„l.662 

,1082 

,1501 

,1920 

,2340 

,2759 

,3178 

420 

1036 

015  3598 

4017 

4436 

4855 

5274 

5693 

6112 

6531 

6950 

7369 

419 

1037 

7788 

8206 

8625 

9044 

9462 

9881 

,0300 

,0718 

,1137 

,1555 

419 

1038 

016  1974 

2392 

2810 

3229 

3647 

4065 

4483 

4901 

5319 

5737 

418 

1039 

6155 

6573 

6991 

7409 

7827 

8245 

8663 

9080 

9498 

9916 

418 

1040 

017  0333 

0751 

1168 

1586 

2003 

2421 

2838 

3256 

3673 

4090 

417 

1041 

4507 

4924 

5342 

5759 

6176 

6593 

7010 

7427 

7844 

8260 

417 

1042 

8677 

9094 

9511 

9927 

,0344 

,0761 

,1177 

,1594 

,2010 

,2427 

417 

1043 

018  2843 

3259 

3676 

4092 

4508 

4925 

5341 

'5757 

'6173 

6589 

416 

1044 

7005 

7421 

7837 

8253 

8669 

9084 

9500 

9916 

,0332 

,0747 

416 

1045 

019  1163 

1578 

1994 

2410 

2825 

3240 

3656 

4071 

4486 

4902 

415 

1046 

5317 

5732 

6147 

6562 

6977 

7392 

7807 

8222 

8637 

9052 

415  1 

1047 

9467 

9882 

»0296 

,0711 

,1126 

,1540 

,1955 

,2369 

,2784 

,3198 

415 

1048 

020  3613 

4027 

4442 

4856 

'5270 

5684 

6099 

6513 

6927 

7341 

414 

1049 

7755 

8169 

8583 

8997 

9411 

9824 

,0238 

,0652 

,1066 

,1479 

414 

1050 

021  1893 

2307 

2720 

3134 

3547 

3961 

4374 

4787 

■5201 

5614 

413 

N. 

L.   0 

1 

2 

3 

4 

» 

' 

7       8 

9 

d. 

LOGAEITHMS  OP  NUMBEKS. 


253 


Table  XXXV, — Containing  hxjariihms  of  numbv.vti  from  1  to  J1,000. — Continued. 
[Extracted  from.  Gauas'  Logarithmic  and  Trigonometric  Tables.] 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

8 

9 

d. 

1050 

021  1893 

2307 

2720 

3134 

3547 

3961 

4374 

4787 

5301 

5614 

413 

1051 

6027 

6440 

6854 

7267 

7680 

8093 

8506 

8919 

9333 

9745 

413 

1052 

022  0157 

0570 

0983 

1396 

1808 

2231 

2634 

3046 

3459 

3871 

413 

1053 

4284 

4696 

5109 

5521 

5933 

6345 

6758 

'  7170 

7583 

7994 

412 

1054 

8406 

8818 

9230 

9643 

,0054 

,0466 

,0878 

,1289 

,1701 

,2113 

412 

1055 

023  2525 

2930 

3348 

3759 

4171 

4583 

4994 

5405 

5817 

6228 

411 

1056 

6639 

7050 

7462 

7873 

8284 

8095 

9106 

9517 

9928 

,0339 

411 

1057 

024  0750 

1161 

1572 

1982 

2393 

2804 

3214 

3025 

4036 

4446 

411 

1058 

4857 

5267 

5678 

6088 

6498 

6909 

7319 

7729 

8139 

8549 

410 

1059 

8960 

9370 

9780 

,0l90 

,0600 

,1010 

,1419 

,1829 

,2239 

,2649 

410 

1060 

025  3059 

3468 

3878 

4288 

4697 

5107 

5516 

5926 

6335 

6744 

410 

1061 

7154 

7563 

7972 

8382 

8791 

9300 

9609 

,0018 

,0427 

,0836 

409 

1062 

026  1245 

1654 

2063 

2472 

2881 

3289 

3698 

4107 

4515 

4924 

409 

1063 

5333 

5741 

6350 

6558 

6967 

7375 

7783 

8192 

8600 

9008 

408 

1064 

9416 

9824 

,0233 

,0641 

,1049 

,1457 

,1865 

,3373 

,2680 

,3088 

408 

1065 

027  3496 

3904 

4312 

4719 

5127 

5535 

5942 

6350 

6757 

7165 

408  1 

1006 

7572 

7979 

8387 

8794 

9201 

9609 

,0016 

,0423 

,0830 

,1237 

407 

1067 

028  1644 

2051 

2458 

2865 

3272 

3679 

4086 

4492 

4899 

5306 

407 

1068 

5713 

6119 

6526 

6932 

73S9 

7745 

8152 

8558 

8964 

9371 

406 

1069 

9777 

,0183 

,0590 

,0996 

,1402 

,1808 

,2214 

,2620 

,3026 

,3433 

406 

1070 

029  3838 

4244 

4649 

5055 

5461 

5867 

6272 

6678 

7084 

7489 

406 

1071 

7895 

8300 

8706 

9111 

9516 

9922 

,0327 

,0732 

,1138 

,1543 

405 

1072 

030  1948 

2353 

2758 

3163 

3568 

3973 

4378 

4783 

5188 

5592 

405 

1073 

5997 

6402 

6807 

7211 

7616 

8020 

8425 

8830 

9234 

9638 

405 

1074 

031  0043 

0447 

0851 

1256 

1660 

2064 

2468 

2872 

3277 

3681 

404 

1075 

4085 

4469 

4893 

5396 

5700 

6104 

6508 

,0Sl7 

7315 

7719 

404 

1076 

8133 

8526 

8930 

9333 

9737 

,0140 

,0544 

,1350 

,1754 

403 

1077 

032  2157 

2560 

2963 

3367 

3770 

4173 

4576 

4979 

5382 

5785 

403 

1078 

6188 

6590 

6993 

7396 

7799 

8201 

8604 

9007 

9409 

9812 

403 

1079 

033  0214 

0617 

1019 

1422 

1824 

2336 

2629 

3031 

3433 

3835 

402 

1080 

4238 

4640 

5042 

5444 

5846 

62  J  8 

6650 

7052 

7453 

7855 

402 

1081 

8257 

8659 

9060 

9462 

9864 

,0265 

,0667 

,1068 

,1470 
5482 

,1871 

402 

1082 

034  2273 

2674 

3075 

3477 

3878 

4279 

4680 

5081 

5884 

401 

1083 

6285 

6686 

7087 

7487 

7888 

8289 

8690 

9091 

9491 

9893 

401 

1084 

035  0293 

0693 

1094 

1495 

1895 

2296 

2696 

3096 

3497 

3897 

400 

1085 

4297 

4698 

5098 

5498 

5898 

6298 

6698 

7098 

7498 

7898 

400 

1086 

8298 

8698 

9098 

9498 

9898 

,0297 

,0697 

,1097 

,^496 

,1896 

400 

1087 

036  2295 

2695 

3094 

3494 

3893 

4393 

4692 

5091 

5491 

5890 

399 

1088 

6289 

6688 

7087 

7486 

7885 

8284 

8683 

9082 

9481 

9880 

399 

1089 

037  0279 

0678 

1076 

1476 

1874 

2272 

2671 

3070 

3468 

3867 

399 

1090 

4265 

4663 

5062 

5460 

5858 

6357 

6655 

7053 

7451 

7849 

398 

1091 

8248 

8646 

9044 

9442 

9839 

,0237 

,0635 

,1033 

,1431 

,1829 

398 

1092 

038  2226 

2624 

3022 

3419 

3817 

4214 

4612 

5009 

5407 

5804 

398 

1093 

6202 

6599 

6996 

7393 

7791 

8188 

8585 

8982 

9379 

9776 

397 

1094 

039  0173 

0570 

0067 

1364 

1761 

2158 

2554 

2951 

3348 

3745 

397 

1095 

4141 

4538 

4934 

5331 

5727 

6124 

6520 

6917 

7313 

7709 

397 

1096 

8106 

8502 

8898 

9294 

9690 

,0086 

,0482 

,0878 

.1274 

,1670 

396 

1097 

040  2066 

2462 

2858 

3354 

3650 

4045 

4441 

4837 

'5232 

5638 

396 

1098 

6023 

6419 

6814 

7310 

7605 

8001 

8396 

8791 

9187 

9582 

395 

1099 

9977 

,0372 

,0767 

,1162 

,1557 

,1953 

,2347 

,3742 

,3137 

,3532 

395 

1100 

N. 

041  3927 
L.   0 

4322 

4716 

5111 

5506 

5900 

6295 

6690 

7084 

8 

7479 

395 

1 

2      3 

4 

5 

6 

9 

d. 

254 


A  MANUAL  OF  TOPOGRArHIC  METHODS. 


Tablic  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents. 
[Extracted  from  Oaiiss'  Logarithmic  anil  Tiigouomctrio  Tables,] 

0° 


' 

L.  Sin. 

(1. 

L.  Tang.  1 

1 

d.  0. 

L.  Cotg. 

L.  Cos. 

0 

1 

0. 00  000 

0. 00  oon 

60 

59 

6.  46  373 

0.46  373 

3.53  627 

6.  76  476 

30103 

6.76  476 

30103    3.23  524 

0.00  000 

58 

17609 

6.94  085 

17609    3.05  915 

0.  00  000 

57 

4 

7.  06  579 

12494 
9691 

7.  06  579 

12494 
9691 

2.93  421 

0.  00  000 

66 

5 

7.16  270 

7. 16  270 

2.83  730 

0.00  000 

55 

6 

7. 24  188 

7918 

7.24  188 

2.75  812 

0.00  000 

7.30  S82 

6691 

7. 30  882 

2.69  118 

0.00  000 

7.36  682 

5800 

7.36  682 

2.63  318 

0.00  000 

52 

11 
10  ^ 

7.41  797 

5115 
4576 

7.41  797 

4576 

2.  58  203 

0. 00  000 

51 

7.  46  373 

7.  46  373 

2.53  627 

0.00  000 

50 

4139 

7.  50  512 

2.49  488 

0.00  000 

49 

7  54  291 

3779 

7.54  291 

2.45  709 

0.  00  000 

48 

7  57  767 

3476 

7.  57  767 

2.42  233 

0.00  000 

47 

14 

7.  60  985 

3218 
2997 

7.  60  986 
7.63  982 

2996 

2.39  014 

0.00  000 

46 

7.63  982 

2.36  018 

0.00  000 

45 

2802 

7. 66  785 

2.  33  215 

0.00  000 

44 

17 

7.69  417 

2633 

7.  69  418 

2482 

2.  30  582 

9.99  999 

43 

7.71  900 

2483 

7. 71  900 

2.28  100 

9.99  999 

19 
20 

7.74  248 

2348 
2227 

7.74  248 

.  2228 

2. 25  752 

9.99  999 

41 

7  76  475 

7. 76  476 

2.  23  524 

9.99  999 

40 

2119 

7.78  595 

2.21  405 

9.99  999 

39 

22 

7.80  615 

2021 

7.80  615 

2. 19  385 

9.99  999 

38 

7.82  545 

1930 

7.82  546 

2. 17  454 

9.99  999 

24 

7.  84  393 

1848 
1773 

7,  84  394 

1773 

2.15  606 

9.99  999 

36 

7.86,166 

7.86  167 

2. 13  833 

9.99  999 

35 

1704 

7.  87  871 

2.12  129 

9.99  999 

34 

27 

7.89  509 

1639 

7.  89  510 

1639 

2. 10  490 

9.99  999 

33 

'S 

7.91  088 

7.91  089 

2.  08  911 

9.99  999 

29 

7.92  612 

1472 

7.  92  613 
7.  94  086 

1473 

2.07  387 

9.99  998 

30 

7.94  084 

2. 05  914 

9.99  998 

30 

7.95  508 

7.95  510 

2.  04  490 

9.99  998 

32 

7.96  887 

1379 

7.96  889 

2.03  111 

9.99  998 

28 

33 

7.98  223 

1336 

7.98  225 

2.01  775 

9.99  998 

27 

34 

7.99  520 

1259 

7.99  522 

1259 

2.  00  478 

9.99  998 

8.08  781 

1.99  219 

9.  99  998 

25 

«  8.  02  0U2 
8.03  192 

1223 

8. 02  004 

1223 

1.97  996 

9.99  998 

24 

1190 

8.03  194 

1.96  806 

9.99  997 

23 

38 

8.04  350 

1158 

8.  04  353 

1.95  647 

9.99  997 

22 

39 

8.05  478 

1100 

8. 05  481 

8. 06  581 

1100 

1.94  519 

9.99  997 

40 

8.06  578 

1.93  419 

9.99  997 

20 

1072 

8.07  653 

1.92  347 

9.99  997 

8.08  696 

1046 

8. 08  700 

1.91  300 

9.99  997 

18 

8.  09  718 

1022 

8.09  722 

1.  90  278 

9.99  997 

17 

44 
45 

8. 10  717 

8.11  693 

999 
976 

8. 10  720 

976 

1.  89  280 

9.99  996 

16 

8. 11  696 

1.88  304 

9.  99  996 

15 

8. 12  651 

1.87  349 

9. 99  996 

14 

8.13  581 

934 

8. 13  585 

934 

1.86  415 

9.  99  996 

13 

48 

8.14  495 

914 

8. 14  500 

915 

1.85  300 

9.  99  996 

12 

49 
SO 

8.15  391 

896 

877 

8. 15  395 

895 

878 

1.84  605 

9.  99  996 

11 

8. 16  268 

8. 16  273 

1.83  727 

9.  99  995 

10 

51 

8.17  128 

8.17  133 

1.82  867 

9.  99  995 

9 

8. 17  971 

8.17  976 

1, 82  024 

9.99  995 

53 

8. 18  798 

827 

8. 18  804 

828 

1.81  196 

9.  99  995 

7 

54 

8. 19  610 

797 

8. 19  616 

797 

1.80  384 

9.99  895 

55 

8. 20  407 

8.  20  413 

1.  79  587 

9.  99  994 

5 

50 

8.21  189 

782 

8.  21  195 

1.78  805 

9.  99  994 

4 

57 

8.21  958 

769 

8. 21  964 

1.78  036 

9.  99  994 

3 

58 

8.22  713 

8.  22  720 

1.77  280 

9.99  994 

59 

8. 23  456 

730 

8.  23  462 
8.  24  192 

730 

1.76  538 
1.  75  808' 

9.99  994 

60 

8.  24  186 

9. 99  993 

0 

L.  Cob. 

d. 

L.  Cotg. 

d.c. 

L.  Tang. 

L.  Sin. 

' 

89= 


LOGAEITHMS  OF  CIECULAE  FUNCTIONS. 


255 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents. — Continued. 

[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

1° 


' 

L.  Sin. 

d. 

L.  Tang. 

d.  0. 

L.  Cotg. 

L.  Cos. 

0 

8.  24  186 

717 
706 
695 
684 
673 

8.  24  192 

718 
706 
696 
684 
673 

1. 75  808 

9.  99  993 

60 

1 

8.24  903 

8.  24  910 

1.  75  090 

9.  99  993 

59 

2 

8.  25  609 

8.25  616 

1. 74  384 

9.99  993 

58 

3 

8. 26  304 

8.  26  312 

1. 73  688 

9.  99  993 

57 

4' 

8. 26  988 

8.  26  S96 

1.73  004 

9. 99  992 

56 

5 

8.27  661 

8.27  669 

1.72  331 

y.99  992 

55 

6 

8.  28  324 

8. 28  332 

1.71  668 

9.99  992 

54 

8. 28  977 

8.  28  986 

1.  71  014 

9.  99  992 

53 

8.  29  021 

8.  29  629 

1.70  371 

9. 99  992 

52 

9 

8. 30  255 

624 

8.30  263 

625 

1.  69  737 

9.  99  991 

51 

10 

8.30  879 

8. 30  888 

1.  69  112 

9.  9'J  991 

50 

11 

8. 31  495 

8. 31  505 

1.  63  495 

9.  99  991 

49 

8.32  103 

8. 32  112 

1.  67  S88 

9. 99  990 

48 

13 

8.  32  702 

8.32  711 

1. 67  289 

9. 99  990 

47 

14 

8.33  292 

583 

8.33  302 

584 

1. 66  698 

9.  99  990 

46 

15 

8. 33  875 

8.33  886 

1.-66  114 

9.99  990 

45 

8.  34  450 

8.  34  461 

1.  65  539 

9.99  989 

44 

17 

8.35  018 

568 

8.  35  029 

568 

1.  64  971 

9.99  989 

43 

8. 35  590 

1.64  410 

9. 99  989 

42 

19 

8.36  131 
8. 36  678 

553 

547 

8.  36  143 

553 
546 

1,  63  857 

9.99  989 

41 

20 

8.  36  689 

1.63  311 

9.99  988 

40 

8.37  217 

8.  37  229 

1. 62  771 

9.  99  988 

39 

2j 

8.37  750 

533 

8.  37  762 

1.  62  238 

9.99  988 

38 

8. 38  289 

1. 61  711 

9.  99  987 

37 

24 

8.38  796 

520 
514 

8.  38  809 

520 
514 

1.61  191 

9.99  987 

36 

25 

8.39  310 

8. 39  323 

1.  60  677 

9. 99  987 

35 

8.39  818 

8.  39  832 

1.  60  168 

9.99  986 

34 

8.40  320 

8.  40  334 

1.  59  660 

9.  99  986 

33 

28 

8.40  816 

8.  40  830 

1.  59  170 

9. 99  986 

32 

29 

8.41  307 

485 

8.41  321 

486 

1.58  679 

9.  99  985 

31 

30 

8.41  792 

8.41  807 

1.58  193 

9.  99  985 

30 

8.42  272 

8.42  287 

1.  57  713 

9.  99  985 

32 

8.42  746 

8.  42  762 

475 

1.57  238 

9.  99  984 

28 

33 

8.43  216 

8.43  232 

1.  56  768 

9.99  984 

27 

34 

8.43  680 

459 

8.43  696 

460 

1. 56  304 

9.  99  984 

26 

8.44  156 

1.65  844 

9.99  983 

25 

36 

8,44  594 

8.44  611 

455 

1.55  389 

9.99  983 

24 

37 

8.45  044 

8.  45  061 

1.54  939 

9. 99  983 

23 

8.  45  489 

8. 45  507 

1. 54  493 

9.99  982 

22 

39 

8.45  930 

436 

8. 45  948 

441 
437 

1.  54  052 

9.  99  982 

21 

40 

8. 46  366 

8. 46  385 

1.53  615 

9. 99  982 

20 

41 

8.46  799 

8.46  817 

1.53  183 

9.  99  981 

19 

8.47  226 

8.47  245 

1. 52  755 

9.99  981 

18 

8.47  650 

424 

8. 47  669 

1.52  331 

9.  99  981 

17 

44 

8.48  069 

419 
416 

8.48  089 

416 

1.51  911 

9.  99  980 

16 

8.48  485 

8.48  505 

1.51  495 

9.  99  980 

15 

46 

8.48  896 

8.48  917 

1.51  083 

9.  99  979 

14 

8.49  304 

8.49  325 

1.  50  675 

9. 99  979 

13 

48 

8.49  708 

404 

8.49  729 

1.50  271 

9.  99  979 

12 

49 

8.  50  108 

396 

8.50  130 

397 

1.49  870 

9.  99  978 

11 

30 

8.50  504 

8.50  527 

1.49  473 

9. 99  978 

10 

51 

8.  50  897 

8.50  920 

1.49  08U 

9.  99  977 

9 

52 

8.51  287 

8.51  310 

1.48  690 

9.99  977 

8 

53 

8.51  673 

386 

8.51  696 

1.48  304 

9. 99  977 

7 

54 

8. 52  055 

379 

8.52  079 

380 

1. 47  921 

9.  99  976 

6 

8.52  459 

1.47  541 

9. 99  976 

5 

8. 52  810 

376 

8.  52  835 

1.47  165 

9.99  975 

4 

57 

8.53  183 

373 

8.  53  208 

1.46  792 

9.  99  975 

3 

58 

8.  53  552 

369 

8.53  578 

1.46  422 

9.  99  974 

2 

59 

8.  53  919 

363 

8. 53  945 
8.  54  308 

363 

1.46  055 

9.  99  974 

1 

60 

8.  54  282 

1.45  692 

9.99  974 

0 

L.  Cos. 

d. 

L.  Cotg. 

d.  c. 

L.  Tang. 

L.  Sin. 

' 

880 


256 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXVI. — Lofiarithmic  sines,  cosines,  tangents,  and  cotangents. — Coutmued. 
[Erfraottd  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


L.  Sin. 

0 

8.  54  282 

8.  54  042 

8.  54  99D 

8.  55  354 

8.55  705 

8.56  054 

8.  56  400 

8.  56  743 

8. 57  084 

8.  o7  421 

10 

8.  57  757 

11 

8.  58  089 

12 

8. 58  419 

13 

8.  58  747 

14 

8.  59  072 

15 

8.  59  395 

16 

8.  .59  715 

17 

8.  60  033 

18 

8.6U  349 

19 

8.  60  662 

20 

8. 60  973 

21 

6.  61  282 

22 

8.61  589 

23 

8.61  894 

24 

8.  62  196 

25 

8.  62  497 

26 

8.  62  795 

27 

8.  63  091 

28 

8.  63  385 

29 
80 

8.63  678 

8. 63  968 

31 

8.  64  256 

32 

8.  64  543 

33 

8.  64  827 

34 

8.65  110 

8.  66  497 

8.66  769 
8.  67  039 

8.67  308 
8.  67  575 

8.67  841 

8. 68  104 
8. 68  367 
8.  68  627 

8. 68  886 

8.69  144 
8.69  400 
8.  69  654 

8.69  907 

8. 70  159 
8.  70  409 
8.  70  658 

8.70  905 

8.71  151 
8.71  395 
8.  71  638 
8.71  880 


87= 


LOGARITHMS  OF  CIRCULAR  FU:srCTIO:SS. 


257 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents — Contiuued. 
[Extracted  from  Gauss'  Logaritttraic  and  Trigonometric  Tables.] 

30 


MON    XXII 


258 


A  MxVNUAL  OF  TOPOGEArHIG  METHODS. 


Table  XXXVI. — Logaritltmic  sines,  cosines,  tanncnts,  and  cotangents — Contiuuod. 
[Bxtractea  from  Gauss'  Losaritlimic  ami  Trigouometric  Tables.] 

4° 


8.  84  358 
S.  84  539 
S.  8i  718 
S.  81  897 
8. 85^075 
8.85^252 
8.  85  429 
8.85  605 
8.  85  780 

8.85  955 
87  86  128 
8.  86  301 

8. 86  474 


8.87  661 

8. 87  829 
8.  87  995 

8. 88  161 
8.88  326 
8.  88  490 
8.  88  654 
8.  88  817 


8.89  142 
8.89  304 
8.  89  464 
8.89  625 
8.  89  784 

8.89  943 

8.90  102 


8.91  807 
8.  91  959 

8.92  110 
8.  92  261 
8.  92  411 


8.  92  561 
8.92  710 

8.92  859 
8.  93  007 
8.  93  154 

8. 93  3DT 
8. 93  448 
8.  93  594 
8.93  740 
8.93  885 
a. 94  030 


8.  84  464 
8.  84  04C 

8. 84  826 

8. 85  006 
8^85  185 

"8.85  363" 
8.  85  540 
8.85  717 

8.85  893 

8.86  069 


8.  86  243 
8.  86  417 
8.  86  591 
8.  86  763 


8.  87  953 
8. 88  120 

8. 88  287 
8.  88  453 
8^88_618^ 

"  8.  88  783 
8.  88  948 

8.89  111 
8.  89  274 

_  8^9^432 
8.  89  598 


8.91  185 
8.91  340 
8.91  495 
8.91  6.50 
8.91  803 


1. 15  536 
1.15  354 
1. 15  174 
1. 14  994 
L14  815 
1. 14  637" 
1. 14  460 
1. 14  283 
1.14  107 
1. 13  931 


1. 13  757 
1. 13  583 
1.13  409 
1.13  237 
1.13  065 
1.12  894 
1.  12  723 
1.12  553 
1.12  384 
1. 12  215 


1.11  880 
1. 11  713 
1. 11  547 
1.11  382 


8.  93  462 
8.93  609 
8.93  756 
8.  93  903 
8.  94  049 
"8.94  195 


L.  Cotg.   d.  c. 


1.11  217 
1. 11  052 
1. 10  889 
1.10  726 
1. 10  563 
1. 10  402 
1. 10  240 
1. 10  080 
1.09  920 
1,09  760 


1.  09  601 
1.09  443 
1.  09  285 
1.09  128 
1.  08  971 
1.  08  815" 
1.  08  060 
1. 08  505 
1. 08  350 
J.  08  197 
f.  08  043" 
1.  07  890 
1.07  738 
1. 07  586 
1. 07  435 


9.  99  894 
9.  99  893 
9.99  892 
9.90  891 
9.99  891  I 
9. 99  990" 
9.  99  889 
9.  99  888 
9. 99  887 


9.  99  880 
9.  99  879 
9.99  879 
9.  99  878 


9.  99  874 
9.  99  873 
9.99  872 
9.  99" 871 
9.  99  870 
9.  99  869 
9.  99  868 


.99  ; 


9.99  861 
9.99  860 
9.  99  859 


9.  99  856 
9.  99  855 
9.  99  854 
9.  99  853 
9. 99  852 


1.  07  284 
1.07  134 


1.06  835 
1.06  687 


1,06  538 
1.06  391 
1. 06  244 
1.  06  097 
JJ)5J)51 
1, 05  805 


9,  99  851 
9,  99  850 
9,99  848 
9,  99  847 
9,  99  846 
9,  99  845 
9, 99  844 
9,  99  Si3 
9.  99  842 
9.  99  841 


9.  99  840 
9.99  839 


9.99  836 
9. 99  834 


182 

ISl 

1V9 

17S 

3,0 

3,0 

3,0 

3,0 

6,1 

6,0 

6,0 

5,9 

9,1 

9,0 

9,0 

8,9 

12,1 

12,1 

11,9 

11,9 

15,2 

15,1 

14,9 

14,8 

18,2 

18,1 

17,9 

17,8 

21,2 

21,1 

20,9 

20,8 

24,3 

24,1 

23,9 

23,7 

27,3 

27,2 

26,8 

26,7 

17G 

175 

174 

17S 

2,9 

2,9 

2,9 

2,9 

5,9 

5,8 

5,8 

5,8 

8,8 

8,8 

8,7 

8,6 

11,7 

11,7 

11,6 

11,5 

14,7 

14,6 

14,5 

14,4 

17,6 

17,5 

17,4 

17,3 

20,5 

20,4 

20,3 

20,2 

23,5 

23,3 

23,2 

23,1 

26,4 

26,2 

26,1 

26,0 

171 

170 

169 

168  ; 

2'8 

2,8 

2,3 

2,8 

5,7 

5,7 

5,6 

5,6 

8,6 

8,5 

8,4 

8,4 

11,4 

U,3 

11,3 

11,2 

14,2 

14,2 

14,1 

14,0 

17,1 

17,0 

16,9 

16,8 

20,0 

19,8 

19,7 

19,6 

22,7 

22,5 

22,4 

25,6 

52'5 

25,4 

25,2 

166 

165 

1G4 

163 

2,8 

2,8 

2,7 

2,7 

5,5 

5,5 

5,5 

6,4 

8,3 

8,2 

8,2 

8,2 

11,1 

11,0 

10,9 

10,9 

13,8 

13,8 

13,7 

13,6 

16,6 

16,5 

16,4 

16,3 

19,4 

19,2 

19,1 

19,0 

22,1 

22,0 

21,9 

21,7 

24,9 

24,8 

24,6 

24,4 

161 

160 

150 

168 

2,7 

2,7 

2,6 

2,6 

5,4 

5,3 

5,3 

6,3 

8,0 

8,0 

8,0 

7,9 

10,7 

10,7 

10,6 

10,5 

13,4 

13,3 

13,2 

13,2 

16,1 

16,0 

15^9 

15,8 

18,8 

18,7 

18,6 

18,4 

21,5 

21,3 

21,2 

21,1 

24,2 

24,0 

23,8 

23,7 

156 

155 

154 

153 

2,6 

2,6 

2,6 

2,0 

5,2 

5,2 

5,1 

5,1 

7,8 

7,8 

7,7 

7,6 

10,4 

10,3 

10,3 

10,2 

13,0 

12,9 

12,8 

12,8 

16,6 

16,5 

15,4 

16,3 

18,2 

18,1 

18'0 

17,8 

20,8 

20,7 

20,5 

20,4 

23,4 

23,2 

23,1 

23,0 

177 

3,0 
6,9 
8,8 
11,8 
14,8 
17,7 
20,6 
23,6 
26,6 

172 

2,9 
6,7 
8,6 
11,5 
14,3 
17,2 
20,1 
22,9 


8,4 
11,1 
13,9 
16,7 
19,5 
22,3 
25,0 


8,1 
10,8 
13,5 
16,3 
18,9 
21,6 
24,3 

157 

2,6 
6,2 
7,8 
10,5 
13,1 
15,7 
18,3 
20,9 
23,6 

152 

2,5 
5,1 
7,6 
10,1 
12,7 
15,2 
17,7 
20,3 
22,8 


85° 


LOGARITHMS  OF  CIECULAE  FUNCTIONS. 


259 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 


91  030 
94  174 
94  317 
94  461 
94  603 


94  746 

94  887 

95  029 
95  170 
95  310 


95  450 
95  589 
95  728 


97  095 
97  229 
97  363 


97  496 
97  629 
97  762 

97  894 

98  026 


98  157 
98  288 
98  419 
98  549 
98  679 


98  937 

99  066 
99  194 
99  332 


99  450 
99  577 
99  704 


00  704 
00  838 

00  951 

01  074 
01  196 
01  318 
01  440 
01  561 
01  682 
01  803 


8.94  195 
S.  94  340 
8. 94  485 
8.  94  630 
8.  94  773 


8.94  917 

8.95  060 
8.  95  202 
8.  95  344 
8.  95  486 


8.95  627 
8.  95  767 

8. 95  908 

8.96  047 
8.  90  187 


8.  96  325 
8.  96  464 
8.96  603 
8.96  739 
8.  96  877 


8.  97  013 
8.97  150 
8. 97  285 
8.  97  421 
8. 97  556 


8.97  691 
8.  97  825 
8.  97  959 
8.  98  092 


8.98  358 
8.  98  490 
8.  98  622 
8.  98  763 


9.  00  301 
9. 00  427 
9.00  553 

9.00  679 
9.  00  805 
9.  00  930 

9.01  055 
9.01  179 
9. 01  303 
9.  01  427 
9.01  550 

9.01  673 
9.  01  796 
9.  01  918 

9.02  040 
9.02  162 


140 
138 


1.  05  805 
1. 05  660 
1.05  515 
1.05  370 
1.  05  227 


1.  05  083 
1.04  940 
1.04  798 
1.04  650 
1.  04  514 


1.  04  373 
1.  04  233 
1.04  092 
1.  03  953 
1.03  813 


1.03  675 
1.  03  536 
1.  03  398 
1. 03  261 
1.03  123 


1. 02  987 
1.02  850 
1. 02  715 
1. 02  579 
1.  02  444 


1.02  309 
1.02  175 
1. 02  041 
1.01  908 
1. 01  775 


1.01  642 
1.01  510 
1.0]  378 
1.01  247 
1.  01  116 


1.  00  985- 
1.  00  855 
1.  00  723 
1. 00  595 
1. 00  466 


1.00  338 
1.  00  209 
1.00  081 
0.99  954 
0  99  826 


99  699 
99  573 
99  447 


98  697 
98  573 
98  450 
98  327 
98  204 
98  C82 
97  960 


824 


9.99  823 
9.  99  822 
9. 99  821 
0.  99  820 
9.  99  819 


9.99  806 
9.  99  81)4 
9.  99  803 
9.  99  802 
9.99  801 


9.  99  800 
9.99  798 
9. 99  797 
9. 99  796 
9.  99  795 


9.99  793 
9.  99  792 
9.  99  791 
9.  99  790 
9. 99  788 


0.99  787 
9.  99  786 
9.99  785 


9. 99  778 
9.  99  777 
9.99  776 


9.  99  771 
:1.  99  769 
9.99  708 
9.  99  767 
9.99  765 
9.99  764 
9.  99  763 
9. 99  761 


L.  Cotg.      d.c.      L.  Tang.  I     L.  Sin. 


149 

148 

2,5 

2,5 

6,0 

4,9 

V,4 

7,4 

9,9 

9,9 

13,4 

12,3 

14,9 

14,8 

r/,4 

17,3 

19,9 

19,7 

22,4 

22,2 

T4+ 

US 

2,4 

2,4 

4,8 

4,8 

V,3 

7,2 

9,6 

9,5 

12,0 

11,9 

14,4 

14,3 

16,8 

16,7 

19,2 

19,1 

31,6 

21,4 

11,2 

11,1" 

13,4 

13,3 

15,6 

15,5 

17,9 

17,7 

30,1 

20,0 

129 

138 

2,2 

2,1 

4,3 

4,3 

6,4 

6,4 

8,6 

8,5 

10,8 

10,7 

12,9 

12,8 

l.'l.O 

14,9 

17,2 

17,1 

19,4 

19,2 

124 

123 

2,1 

2,0 

4,i 

•t,l 

6,2 

6,2 

8,3 

8,2 

10,3 

10,2 

13,4 

12,3 

14„'> 

14,4 

16,5 

16,4 

18,6 

18,4 

4,1 
6,1 
8,1 
10,2 
12,2 
14,2 
16,3 
18,3 


146 

2,4 
4,9 
7,3 
9,7 
12,2 
14,6 
17,0 
19,5 
21,9 
141 
2,4 
4,7 
7,0 
9,4 
11,8 
14,1 
16,4 
18,8 
21,2 
136 
2,3 
4,5 
6,8 
9,1 
11,3 
13,6 
15,9 
18,1 
20,4 

131 

2,2 


8,7 
10,9 
13,1 
15,3 
17,5 
19,6 

126 

2,1 
4,2 
6,3 
8,4 
10,5 
13,6 
14,7 
16,8 
18,9 
121 


10,1 
12,1 
14,1 
16,1 
18,2 


84= 


2(30 


A  MA]S^UAL  OF  TOPOGEAPUIC  METHODS. 


Taulk  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cottingenls — Contiuued. 
[Extractu<Uro:n  Gauss'  Logarithmic  anil  Trigouometric  Tables.] 


121 

120 

119 

2,0 

2,0 

2,0 

4,(1 

4,0 

4,0 

(i,0 

0,0 

6,0 

8,1 

8,0 

7,9 

10,1 

10,0 

9,9 

12,1 

12,0 

11,9 

14,1 

14,0 

13,9 

1li,l 

16,0 

15,9 

1H,2 

18,0 

17,8 

20,2 

20,0 

19,8 

4o,;i 

40,0 

39,7 

60,5 

G0,0 

59,5 

80,7 

80,0 

79,3 

100,8 

100,0 

99,2 

117 

110 

115 

2,0 

1,9 

1,9 

a,9 

3,9 

3,8 

5,8 

6'8 

5,8 

V,8 

7,7 

7,7 

9,8 

9,7 

9,6 

11,7 

11,6 

11,5 

13,6 

13,5 

13,4 

lo,B 

15,5 

15,3 

IV, (i 

17,4 

17,2. 

19,5 

19,3 

19,2 

39,0 

38,7 

38,3 

i)8,b 

58,0 

57,5 

V8,0 

77,3 

76,7 

9V,o 

96,7 

95,8 

lis 

112 

111 

1 

1,9 

1,9 

1,8 

2 

3,8 

3,7 

3,7 

3 

5,6 

5,6 

5,6 

4 

7,5 

7,5 

7,4 

.■i 

9,4 

9,3 

9,2 

(i 

11,3 

11,2 

11,1 

V 

13,2 

13,1 

13,0 

8 

15,1 

14,9 

14,8 

9 

17,0 

16,8 

16,6 

10 

18,8 

18,7 

18,5 

20 

37,7 

.37,8 

37,0 

30 

56,5 

56,0 

55,5 

40 

75,3 

74,7 

74,0 

60 

94,2 

93,3 

92,5 

109 

108 

107 

1,8 

1,8 

1,8 

3,6 

3,6 

3,6 

5,4 

5,4 

5,4 

7,3. 

7,2 

7,1 

9,1 

9,0 

8,9 

10,9 

10,8 

10,7 

12,7 

12,6 

12,5 

14,5 

14,4 

14,3 

16,4 

16,2 

16,0 

18,2 

18,0 

17,8 

36,3 

36,0 

35,7 

54,5 

54,0 

53,5 

72,7 

72,0 

71,3 

90,8 

90,0 

89,2 

118 

2,0 


15,7 
17,7 
19,7 
39,3 


114 

1,9 


9,5 
11,4 
13,3 
15,2 
17,1 
19,0 
38,0 
57,0 
76,0 
95,0 


110 

1,8 
3,7 
5,5 
7,3 
9,2 
11,0 
12,8 
14,7 
16,5 
18,3 
36,7 
55,0 
73,3 
91,7 


106 

1,8 
3,5 
5,3 


10,6 
12,4 
14,1 
15,9 
17,7 
35,3 
53,0 
70,7 


LOGARITHMS  OF  OIRCULAE  FUNCTIONS. 


261 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  froniGanss'  Logarithmic  aiul  Trigonometric  Tables.] 


1.   L.  Tang. 


I  999 


9.09  101 
9.09  202 
9.  09  304, 
9.  09  405 
9. 09  506 


9.09  606 
9.  09  707 
9.09  807 
9.  09  907 
0. 10  000 


9. 10  106 
9. 10  20.5 
9. 10  304 
9.10  402 
9. 10  501 


9.10  599 
9. 10  697 
9.10  795 
9.  10  893 
9. 10  990 


9.11  087 
9. 11  184 
9.11  281 
9. 11  377 
9. 11  474 
9. H  570 
9. 11  666 
9.11  761 
9.11  857 

9.11  953 
S.  12  047 

9. 12  142 
9. 12  236 
9. 12  331 
9. 12  425 
9. 12  519 
9. 12  012 
9.12  706 
9. 12  799 
9.12  892 

9. 12  985 

9.13  078 
9.13  171 
9. 13  263 
9. 13  355 
9. 13  447 
9.13  539 
9.13  630 
9. 13  722 
9. 13  813 
9.  13  904 

9.13  994 

9. 14  085 
9. 14  175 
9. 14  266 
9. 14  856 


92 
91 


9.  08  914 
9.  09  019 
9.09  123 
9.09  227 
9.  09  330 


n. 09  434 
9.  09  537 
9.09  640 
9.  09  742 
9.09  845 


9. 09  947 

9.10  049 
9.10  150 
9.10  253 
9. 10  353 


9.10  J54 
9.10  5.55 
9. 10  056 
9. 10  756 
9.10  850 


9.10  956 

9.11  O.iO 
9.11  155 
9. 11  254 
9.11  353 


9.11  452 
9.11  551 
9. 11  G49 
9. 11  747 
9. 11  845 

9. 11  943 

9. 12  040 
9. 12  138 
9. 12  235 
9. 12  332 
9. 12  428 
9. 12  625 
9. 12  621 
9. 12  717 
9. 12  813 

9. 12  909 
9.  13  004 

9. 13  099 
9.13  194 
9. 13  289 
9.13  384 
9. 13  478 
9. 13  573 
9. 13  667 
9. 13  761 
9. 13  854 

9. 13  948 

9. 14  041 
9. 14  134 
9. 14  227 
9. 14  320 
9. 14  412 
9. 14  504 
9.  14  597 
9. 14  688 
9. 14  780 


0.91  080 
0.  90  981 
0.90  877 
0. 90  773 
0.90  670 


0.90  053 
0.  89  951 
0.  89  850 
0.  89  748 


9. 99  675 
9.  99  674 
9.  99  672 
9. 99  670 
9.  99  669 


9.99  667 
9. 99  666 
9. 99  664 
9. 99  663 
9.99  661 


9.99  659 
9.99  658 
9. 99  656 
9.99  655 
9.99  653 


0.89  244 


0.1 


144 


9.99  651 
9.  99  650 
9.99  64S 
9.  99  647 
9, 99  645 


0.69  044 
0.  88  944 
0.  88  845 
0. 88  746 
0.  88  647 


9.99  643 
9.  99  642 
9.99  640 
9.  99  638 
9.99  637 


0.83  548 
0.  88  449 
0.88  351 
0.  83  253 
0.  88  155 
0. 88  057  I 
0.87  960 
0. 87  862 
0.  87  765 
0. 87  068  I 
0. 87  572 
0.  87  475 
0.  87  379 
0.87  283 
0.  87  187 
0.87  U9l 
0.86  I 
0.  86  901 
0.86  80G 
0.86  711 
0.86  016 
0.86  522 
0.  86  427 
0.  86  333 
0.  86  239 
0.86  146 
0.86  052 
0.  85  959 
0.85  866 
0. 85  773 
0.85  680 
0.1 
0.  85  496 
0.85  403 
0.  85  312 


9.  99  635 
9.  99  633 
9.99  632 
9.  99  630 
9.  99  629 
9.99  627 
9.  99  625 
9.99  624 
9. 99  622 
9.  99  620 
9.  99  618  I 
9.  99  617 
9.99  615 
9.  99  613 
9.99  r  ' 
9.99  610  1 
9.  99  608 
9.  99  607 
9.  99  605 

9.  99  603  I 

9799  601  1   15 


9.! 


I  600 


9. 99  586 
9.  99  584 
9.  99  582 
9.99  581 
9.  99  579 
9. 99  577 
9.  99  575 


10.1 

104 

103 

102 

1,8 

1,7 

1,7 

1,7 

3,5 

3,5 

3,4 

3,4 

5,2 

5,3 

5,3 

5,1 

V,0 

6,9 

6,9 

0,8 

K,K 

8,7 

8,6 

8,5 

10,5 

10,4 

10,3 

10,2 

12,3 

13,1 

12,0 

11,9 

14,0 

13,9 

13,7 

13,0 

15,8 

15,6 

15,4 

15,3 

17,5 

17,3 

17,3 

17,0 

35,0 

34,7 

34,3 

34,0 

52,5 

53,0 

51,5 

51,0 

70,0 

69,3 

68,7 

68,0 

8V,b 

86,7 

85,8 

85,0 

101 

100 

99 

98 

1,V 

1,7 

1,6 

1,6 

3,4 

3,3 

3,3 

3,3 

5,0 

5,0 

5,0 

4,9 

6,7 

6,7 

6,6 

6,5 

8,4 

8,3 

8,2 

8,2 

10,1 

10,0 

9,9 

9,8 

11,8 

11,7 

11,6 

11,4 

13,  b 

13,3 

13,2 

13,1 

15,2 

15,0 

14,8 

14,7 

16,8 

16,7 

16,5 

16,3 

3H,V 

33,3 

33,0 

32,7 

bO,b 

50,0 

49,5 

49,0 

BV,3 

■66,7 

66,0 

65,3 

84,2 

83,3 

83,5 

81,7 

97 

96 

95 

94 

1,6 

1,6 

1,6 

1,6 

3,2 

3,2 

3,2 

3,1 

4,8 

4,8 

.  4,8 

4,7 

6,5 

6,4 

6,3 

6,3 

8,1 

8,0 

7,9 

7,8 

9,7 

9,6 

9,5 

9,4 

11,3 

11,2 

11,1 

11,0 

13,9 

12,8 

12,7 

12,5 

14,6 

14,4 

14,2 

14,1 

16,2 

16,0 

15,8 

15,7 

33,3 

32,0 

31,7 

31,3 

48,b 

48,0 

47,5 

47,0 

64,7 

64,0 

63,3 

62,7 

80,8 

80,0 

79,2 

78,3 

OS 

92 

91 

90 

1,6 

1,5 

1,5 

1,5 

3,1 

3,1 

3,0 

3,0 

4,6 

4,6 

4,6 

4,5 

6,2 

6,1 

6,1 

6,0 

7,8 

7,7 

7,6 

7,5 

9,3 

9,2 

9,1 

9,0 

10,8 

10,7 

10,6 

10,5 

12,4 

13,3 

12,1 

12,0 

14,0 

13,8 

,  13,6 

13,5 

ib,b 

15,3 

15,2 

15,0 

31,0 

30,7 

30,3 

30,0 

46,5 

46,0 

45,5 

45,0 

62,0 

61,3 

60,7 

60,0 

7V,6 

76,7 

75,8 

75,0 

83= 


262 


A  MANUAL  OF  TOPOaKAPHIC  METHODS. 


Table  XXXVI. — Logarithmic  shies,  cosines,  tangents,  and  cotangents — Contiuned. 
[Extracted  from  Gauss'  Logaritlimic  and  Trigonometvic.  Tables.] 


9. 14  356 
9.U  445 
9. 14  535 
9. 14  624 
9. 14  714 


9. 14  8U3 
9.  14  891 

9. 14  980 

9. 15  069 
9. 15  157 


9. 15  245 

9. 15  333 

9. 15  421 

9.15  508 

9. 15  596 


9.16  116 
9.  16  203 
9. 16  289 
9. 16  374 
9. 16  460 


9. 16  545 
9. 16  631 
9. 16  716 
9. 16  801 
9. 16  886 


9. 16  970  I 

9. 17  055  1 
9. 17  139 
9. 17  223 
9. 17  307 
6. 17  391 
9. 17  474 
9.17  ,558 
9. 17  641 


9. 17  973 

9. 18  055 
9. 18  137 
9. 18  220 
9. 18  302 
9. 18  383  ! 
9. 18  465  i 
9. 18  547  { 
9.18  628 
9.18  709 
9.18  790 
9.18  871 

9. 18  952 

9. 19  033 
9.19  113 
9.19  193 
9. 19  273 
9. 19  353 
9.19  433 


9. 14  780 
9. 14  872 

9. 14  963 

9. 15  054 
9.15  145 


9. 15  236 
9. 15  327 
9.15  417 
9. 15  508 
9. 15  598 


9. 17  965 

9.18  051 
9.18  136 
9. 18  221 
9. 18  306 
9.18  391 
9. 18  475 
9.18  560 
9. 18  644 
9.18  728 
9. 18  812 
9. 18  896 

9. 18  979 

9. 19  063 
9. 19  146 
9.19  229 
9.19  312 
9.19  395 
9. 19  478 
9. 19  561 
9.19  643 
9. 19  725 
9.19  807 
9. 19  889 
9. 19  971 


0.  85  220 
0.85  128 
0.85  037 
0.84  946 
0.84  855 


0.84  764 
0.84  673 
0.S4  583 
0. 84  492 
0.84  402 


0.84  312 
0.84  223 
0.  84  133 
0.84  044 
0.  83  954 


0.  83  865 
0.  83  776 
0.83  688 
0.83  599 
0.83  511 


0.83  423 
0.83  335 
0.  83  247 
0.83  159 
0.83  072 


0.  82  984 
0.82  897 
0.82  810 
0. 82  723 
0.  82  637 


0.82  550 
0.  82  464 
0.82  378 
0.  82  292 
0.82  201; 
0. 82  120 
0.82  035 
0.81  949 
Q.81  864 
0.81  779 
0.81  694 
0.81  609 
0.81  525 
0.  81  440 
0. 81  356  I 
0. 81  272  I 
0.  81  188 
0.  81  104 
0.81  021 
0.  80  937 


9.99  572 
9.99  570 
9.99  568 


9. 99  566 
9.99  565 
9.99  563 
9.99  561 
9.99  559 


9.99  557 
9. 90  556 
9.99  554 
9.99  5.52 
9.99  550 


9. 99  548 
9. 99  546 
9.99  545 
9. 99  543 
9.99  541 


9.99  528 
9.99  526 
9.99  524 
9.99  522 


9.99  520 
9.  99  518 
9.  99  517 
9.99  515 
9.99  513 
9.99  511 
9.99  509 
9.99  507 
9.99  505 
9.99  503 
9.99  501 
9.99  499 
9.99  497 
9.99  495 
9.99  494 
9.99  492 
9.99  490 
9.99  488 
9.  99  486 
9.99  484 
9.99  482 
9.99  480 
9.99  478 
9.  99  476 
9.99  474 
9.  99  472 
9.99  470 
9.99  468 
9.  99  466 
9.99  464 
9.  99  462 


L.  Tang. 


92 

91 

90 

1,5 

1,5 

1,5 

3,1 

3,0 

3,0 

4,6 

4,6 

4,5 

•  6'1 

6,1 

6,0 

7/7 

7,6 

7,5 

9,2 

9,1 

9,0 

10,7 

10,6 

10,5 

12,3 

12,1 

12,0 

13,8 

13,6 

13,5 

15,3 

15,2 

15,0 

30,7 

30,3 

30,0 

46,0 

45,5 

45,0 

61,3 

60,7 

60,0 

76,7 

75,8 

75,0 

S9 

88 

8J 

1,5 

1,5 

1,4 

3,0 

2,9 

2,9 

4,4 

4,4 

4,4 

5,9 

5,8 

7,4 

7,3 

7,2 

8,9 

8,8 

8,7 

10,4 

10,3 

10,2 

11,7 

.11,6 

13,4 

13,2 

13,0 

14,8 

14,7 

14,5 

29,3 

29,0 

44,0 

43,5 

59,3 

58,7 

58,0 

V4,2 

73,3 

72,5 

86 

85 

84 

1,4 

1,4 

1,4 

2,8 

2,8 

4,3 

4,2 

4,2 

0,1 

5,7 

5,6 

'1,'i 

7,1 

7,0 

8,b 

8,5 

«8,4 

10,0 

9,9 

9,8 

ll,b 

11,3 

11,2 

12,9 

12,8 

12,6 

14,3 

14,2 

14,0 

28,7 

28,3 

28,0 

43,0 

42,5 

42,0 

57,3 

56,7 

56,0 

71,7 

70,8 

70,0 

■S» 

82 

81 

1,4 

1,4 

1,4 

2,7 

2,7 

4,2 

4,1 

4,0 

5,5 

6,5 

5,1 

6,9 

6,8 

6,8 

8,3 

8,2 

8,1 

9,6 

9,4 

11,1 

10,9 

10,8 

12,4 

12,3 

12,2 

13,8 

13,7 

13,5 

27,7 

27,3 

27,0 

41,5 

41,0 

40,5 

55,3 

54,7 

54,0 

69,2 

68,3 

67,5 

81^ 


LOGAEITHMS  OF  CIRCULAR  FUNCTIONS. 


263 


Table  XXXVI. — Loganthmic 
[Extracted  ftom  Gai 


sines,  cosines,  tangents,  and  cotangents — Continued, 
iss'  Logaritlunic  and  Trigonometric  Tables.] 


9.19  433 
9. 19  613 
9.19  592 
9. 19  672 
9. 19  751 


9.19  830 
9.19  909 

9. 19  988 
9.  20  067 

9.20  145 


9.20  223 
9.  20  302 
9.20  380 
9.20  458 
9. 20  535 


9.20  768 
9.  20  846 
9^20  922^ 

9.20  999 
9.  21  076 

9.21  153 
9.21  229 
9.21  306 


9.21  382 
9.21  468 
9.21  534 
9.21  610 
9^21_685^ 
9.21  761 
9.21  836 
9.21  912 
9. 21  987 
9.  22  002 


9.  22  137 
9.  22  211 
9.  22  286 
9.22  361 
9. 22  435 


9.  22  509 
9,  22  583 
9.  22  657 
9.22  731 
9.22  805 


9.22  952 

9.23  025 
9.23  098 
9.23  171 
9.  23  244 
9.23  317 
9.23  390 
9. 23  463 
9.23  535 
9.23  607 
9.23  679 
9.23  752 
9.23  823 
9. 23  895 
9.  23  967 


9. 19  971 
9.  20  053 
9.  20  134 

9.20  216 
9.20  297 


9.20  378 
9.20  469 
9.  20  540 
9.  20  021 
9. 20  701 


9.  22  747 
9.22  824 
9.22  901 

9.22  977 

9.23  054 
9.23  130 
9.23  206 
9.23  283 
9.  23  359 
9.  23  435 


9.  23  510 
6.23  586 
9.23  661 
9.23  737 
9.  23  812 
9.23  887 

9. 23  962 

9.24  037 
9.24  112 
9.24  186 
9.  24  261 
9.24  335 
9.  24  410 
9.  24  484 
9.  24  558 
9. 24  632 


0.  79  947 
0.  79  866 
0.79  784 
0.79  703 


0.  79  622 
0.  79  541 
0.79  460 
0.  79  379 
0. 79  299 


0.  79  218 
0.79  138 
0.79  058 
0.  78  978 


0.  78  739 
0. 78  659 
0.78  580 
0. 78  501 


0.  78  422 
0. 78  343 
0.78  264 
0.78  186 
0.78  107 


0.78  029 
0. 77  951 
0. 77  873 
0.77  795 
0.77  717 


9. 99  462 
9. 99  460 
9.99  458 
9.99  456 
9. 99  454 


9.99  442 
9. 99  440 
9.99  438 
9.99  436 
9. 99  434 


9,  99  432 
9.99  429 
9. 99  427 
9.99  425 
9.  99  423 


9.99  421 
9.99  419 
9.  99  417 
9.99  415 
9.99  413 


9.99  411 
9.99  409 
9.  99  407 
9.99  404 
9.  99  402 


9.99  396 
9. 99  394 
9.  99  392 


0.  76  113 
0.  76  038 
0.75  963 
0.  75  888 
0. 75  814 
0. 75  739  I 
0.75  665 
0.75  590 
0.75  516 
0.75  442 
0.751 


L.  Tang. 


9. 99  390 
9. 99  388 
9. 99  385 
9.99  383 
9.  99  381 
9.  99  379 
9. 99  377 
9.  99  375 
9.  99  372 
9. 99  370 
9.99  368 
9.99  366 
9. 99  364 
9.99  363 
9.  99  359 
9. 99  357 
9. 99  355 
9.  99  353 
9.  99  351 
9.  99  348 
9.  99  346 
9.99  344 
9. 99  342 
9. 99  340 
9. 99  337 
9. 99  335 


80 

79 

78 

1,3 

1,3 

1,3 

2,7 

2,6 

2,6 

4,0 

4,0 

3,9 

5,3 

5,3 

5,2 

6,7 

6,6 

6,5 

8,0 

7,9 

7,8 

9,3 

9,2 

9,1 

10,7 

10,5 

10,4 

12,0 

11,8 

11,7 

13,3 

13,2 

13,0 

26,7 

26,3 

26,0 

40,0 

39,5 

39,0 

53,3 

52,7 

52,0 

66,7 

65,8 

65,0 

76 

75 

74 

1,3 

1,2 

1,2 

2,5 

2,5 

2,5 

3,8 

3,8 

3,7 

5,1 

5,0 

4,9 

6,3 

6,2 

6,2 

7,6 

7,5 

7,4 

8,9 

8,8 

8,6 

10,1 

10,0 

9,9 

11,4 

11,2 

11,1 

12,7 

12,5 

12,3 

25,3 

35,0 

24,7 

38,0 

37,5 

37,0 

50,7 

50,0 

49,3 

63,3 

62,5 

61,7 

72 

71 

S 

1,2 

1,2 

0,0 

0,1 

3,6 

3,6 

0,2 

4,8 

4,7 

0,2 

6,0 

5,9 

0,2 

7,2 

7,1 

0,3 

8,4 

8,3 

0,4 

9,6 

9,5 

0,4 

10,8 

10,6 

0,4 

12,0 

11,8 

0,5 

24,0 

23,7 

1,0 

36,0 

35,5 

1,5 

48,0 

47,3 

2,0 

60,0 

59,2 

2,5 

3 

79 

3 

78 

13,2 
39,5 
65,8 

13,0 
39,0 
65,0 

12,5 
37,5 
62,5 


12,8 
38,5 
64,3 


13,3 
37,0 
61,7 


9,7 
11,0 
13,2 
24,3 


80° 


264 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXXVI. — Logaritnmic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

10° 


9. 23  967 

9.24  039 
9.  24  110 
9.24  181 
9.24  253 


9.  24  324 
9.24  395 
9.  24  466 
9.24  536 
9.  24  607 


9.25  098 
9.25  168 
9. 25  237 
9.25  307 


9.25  721 
9.  25  790 
9.  25  858 
9.25  927 
9.  25  995 


9. 26  063 
9. -6  131 
9.26  199 
9. 26  267 
9.26  335 


9.  26  403 
9.  26  470 
9.  26  538 
9. 26  605 
9.  26  672 


9.  26  739 
9.  26  806 
9.  26  873 
9.  26  940 
9.27  207 


9. 27  073 
9.27  140 
9.  27  206 
9. 27  273 
9. 27  339 


9. 27  405 
9.  27  471 
9.  27  537 
9.  27  602 
9.  27  668 


9.  27  734 
9.  27  799 
9.  27  864 
9.  27  930 
9.  27  995 


9.  24  632 
9.  24  706 
9.  24  779 
9.  24  853 
9.  24  926 


I.  25  000 
'.25  073 


9.25  365 
9.25  437 
9.25  510 
9.  25  582 
9.25  655 


9.25  727 
9.  25  799 
9.  25  871 
9. 25  943 
9.  26  015 


9.  26  086 
9.26  158 
9.  26  229 
9. 26  301 
9.  20  372 


9.20  443 
9.  26  514 
9.  26  585 
9.26  655 
9.  26  726 


9.  26  797 
9.  26  867 

9. 26  937 

9.27  008 
9.  27  078 


9.27  148 
9.27  218 
9. 27  288 
9.27  357 
9. 27  427 


9.  27  496 
9.  27  566 
9.  27  635 
9.27  704 


9.28  254 
9.28  323 
9.  28  391 
9.  28  459 


9.  28  527 
9. 28  595 
9.28  662 
9.  28  730 
9.28  798 
9.28  865 


0.  74  635 
0.  74  563 
0. 74  490 
0.74  418 
0.  74  345 


0.  74  273 
0. 74  201 
0.  74  129 
0.  74  057 
0.  73  985 


0.73  914 
0.73  842 
0.  73  771 
0.73  699 
0.  73  628 


0. 73  657 
0.  73  486 
0.  73  415 
0.  73  345 
0  73  274 


0.  73  203 
0.73  133 
0.  73  063 
0.  72  992 
0.  72  922 


0. ' 


I  852 


0.72 
0.72  712 
0. 72  643 
0.  72  573 


0.  72  504 
0.  72  434 
0.  72  365 
0.72  296 
0. 72  227 
0.72T5¥ 
0.  72  089 
0. 72  020 
0.7]  951 
0. 71  883 


0. 71  814 
0. 71  746 
0.  71  677 
0. 71  609 
0.  71  541 


L.  Tang. 


9.99  331 
9.  99  328 
9.  99  326 


9.  99  324 
9.  99  322 
9.99  319 
9.99  317 
9.  99  315 


9.  99  301 
9. 99  299 
9.99  297 
9.  99  294 
9. 99  292 


9.99  290 
9.  99  2S8 
9.99  285 
9.99  283 
9.99  281 


9.99  270 
9.  99  274 
9.  99  271 
9.99  269 
9. 99  267 
9.  99  264 
9.  99  202 
9.  99  260 
9.  99  257 


9.  99  219 
9.  99  217 
9.  99  214 


9.  99  200 
9.  99  197 
9.  99  195 


74 

73 

1,2 

1,2 

2,5 

2,4 

3,7 

3,6 

4,9 

4,9 

6,2 

0,1 

7,4 

7,3 

8,6 

8,5 

9,9 

9,7 

11,1 

11,0 

12,3 

12,2 

24,7 

24,3 

37,0 

36,5 

49,3 

48,7 

61,7 

60,8 

71 

1,2 

70 

1,2 

2,4 

2,3 

3'6 

3,5 

4,7 

4,7 

5,9 

5'8 

7,1 

7,0 

8,3 

8,2 

9,5 

9,3 

10,6 

10,5 

11,8 

11,7 

23,7 

23,3 

35,5 

35,0 

47,3 

46,7 

59,2 

58,3 

08 

67 

1/1 

1,1 

2,3 

2,2 

3,4 

3,4 

4,5 

4,5 

5,7 

5,6 

6,8 

6,7 

7,9 

7,8 

9,1 

8,9 

10,2 

10,0 

11,3 

11,2 

22,7 

22,3 

34,0 

33,5 

45,3 

44,7 

56,7 

55,8 

9,0 

10,8 
12,0 
24,0 
30,0 
48,0 
00,0 


3,4 
4,6 
5,8 
6,9 
8,0 
9,2 
10,4 
11,5 
33,0 
34,5 
46,0 
E.7,5 

66 

1,1 
2,2 
3,3 
4,4 
5,5 
6,6 


9,9 
11,0 
22,0 
33,0 
44,0 
53,0 


12,3  12,2  12,0 
.37,0  36,5  36,0 
61,7  i  60,8   60,0 


3 
71 

3 

JO 

3 
69 

11,8 
35,5 
59,2 

11,7 
35,0 
58,3 

11,5 
34,5 
57,5 

11,3 
34,0 
56,7 


79° 


LOGAEITHMS  OF  CmCULAR  FUNCTIONS. 


265 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents- 
[Extracted  from  Gauss' Logarithmic  and  Trigonometric  Tables.] 

11° 


9.28  060 
9.28  125 
9.  28  190  I 
9.28  254 
9.28  319 


9  28  705 
9.28  769 
9.28  833 
9.28  896 
9.28  960 


9.29  024 
9.29  087 
9.29  150 
9.29  214 
9.29  277 


9.29  340 
9.  29  403 
9.29  466 
9.29  .529 
9.29  591 


9.29  054 
9.29  716 
9,29  779 
9.29  841 
9.29  903 


9.29  966 

9.30  028 
9.30  090 
9.  30  151 
9.  30  213 


9.30  887 

9.30  947 

9.31  008 
9.31  068 
9.31  129 
9.31  i89 
9.31  250 
9.31  310 
9.31  370 
9.  31  430 
i).  31  490 
9.31  549 
9.31  609 
9.31  069 
9.31  728 
9:31  788 


9.28  384 
9.28  448  i 
9.28  512 
9.28  577 
9.28  641  I 


9.28  865 
9.  28  933 

9.29  000 
9.29  067 
9.29  134 


9.29  201 
9.29  268 
9.29  335 
9.29  402 
9. 29  468 


9.29  535 
9.29  601 
9.29  668 
9.29  734 
9.29  800 


9. 39  932 

9.29  998 

9.30  064 
9.30  130 
9.30  195 
9.30  261 
9.30  326 
9.30  391 
9.  30  457 


9.30  522 
9.30  587 
9.30  652 
9.30  717 
9.30  782 


9.  30  846 
9.30  911 

9.30  975 

9.31  040 
9.31  104 


9.31  168 
9.31  233 
9.31  297 
9.31  361 
9.  31  425 
9.31  489^ 
9.31  552 
9.31  616 
9.31  679 
9.31  743 
9. 31  806 
9.31  870 
9.31  933 

9.31  996 

9.32  059 


9.  32  436 
9.32  498 
9.  32  561 
9.32  623 
9.32  685 
9.  32  747 


L.  Cotg. 


L.  Cotg. 


0.71  135 
0.71  067 
0.71  000 
0.70  933 
0,  70  866 
0.70  799 
0.70  732 
0.70  005 
0.70  598 
0.  70  532 


0.  70  465 
0.  70  399 
0.70  332 
0.70  266 
0.70  200 


0.70  134 
0.70  068 
0.70  002 
0.69  936 
0.69  870 
0.  69^80"5~l 
0.  09  739  I 
0.  69  674 


9.99  190 
9.99  187 
9.99  185 


9.99  170 
9.99  167 
9.99  165 
9.99  162 
9.99  160 


9.99  157 
9.99  155 
9.99  152 
9.99  150 
9.  99  147 
9.  99  145 
9.99  142 
9.99  140 


0.  69  413 
0.69  348 
0,  69  283 
0. 69  218 


0.68  767 
0.68  703 
0.68  639 


0^68  2J)7 
0.  68  194 


JUi7  941_ 
0.67  878 
0. 67  815 
0.  67  7.52  I 
0.  67  689 

_a  W  627^ 
0.  67  564 
0.  67  502  1 
0. 67  439 
0.67  37 
0.  07  315 


9.99  112 
9.99  109 


9.99  106 
9.99  104 
9.99  101 
9.99  099 
9.09  096 
9. 99  093 
9.99  091 
9.99  088 
9.99  086 
9.99  083 


9.  99  062 
9.99  059 
9.99  056 
9.99  054 
9.99  051 
9.99  048 
9.99  046 
9.99  043 
9. 99  040 


65 

Gi 

1/1 

1,1 

2,2 

2,1 

3,2 

3,2 

4,3 

4,3 

5,4 

5,3 

6,5 

0,4 

7,6 

7,5 

8,7 

8,5 

9,8 

9,6 

10,8 

10,7 

21,7 

21,3 

32,5 

32,0 

43,3 

42,7 

54,2 

53,3 

62 

61 

1,0 

1,0 

2,1 

2,0 

3,1 

3,0 

4,1 

4,1 

5,2 

5,1 

6,2 

6,1 

7,2 

7,1 

8,3 

8,1 

9,3 

9,2 

10,3 

10,2 

2C,7 

20,3 

31,0 

30,5 

41,3 

40,7 

51,7 

50,8 

59 

1,0 

3 

0,0 

2,0 

0,1 

3,0 

0,2 

3,9 

0,2 

4,8 

0,2 

5,9 

a       0,3 

6,9 

0,4 

7,9 

0,4 

8,8 

0,4 

9,8 

0,5 

19,7 

1,0 

29,5 

1,5 

39,3 

2,0 

49,2 

2,5 

10,7 
32,0 
53,3 


11,0 
33,0 
56,0 


10,5 
31,5 
52,5 


1,0 
2,1 
3,2 
4,2 

6^3 
7,4 
8,4 
9,4 
10,5 
21,0 
31,5 
42,0 
52,5 


10,0 
20,0 
30,0 
40,0 
50,0 


10,8 
32,5 
54,2 


78^ 


266 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXyi.—Loijayithmic 
[Extractocl  fnim  Giii 


incs,  cosines,  tangents,  and  cotangents — Continued. 
as'  Logarithmic  and  Trigouomotric  Tables.] 


9.31  788 
9.31  847 
9.31  907 
9.31  9CC 


9.32  202 
9.32  2G1 
9.32  319 


9.32  437 
9.32  49.5 
9.32  553 


9.32  S44 
9.32  902 


9.32  960 

9.33  018 
9.33  075 
9.33  133 
9.33  190 


9.33  248 
9.33  305 
9.33  362 
9.33  420 
9.33  477 


9.33  761 


9.33  818 
9.33  874 
9.33  831 

9.33  987 

9.34  043 


•55 


9.34  100 
9.34  156 
9.34  212 
9.34  268 
9.34  324 


9.34  380 
9.34  436 
9.34  491 
9.34  547 
9.34  602 


9.34  058 
9.34  713 
9.34  769 
9.34  824 
9.34  879 


9.34  934 

9.34  989 

9.35  044 


9.32  933 
9.32  995 


9.33  057 
9.33  119 
9.33  180 
9.33  242 
9.33  303 
"9.33  365 
9.33  426 
9.33  487 


9.33  670 
9.33  731 
0.33  792 
9.33  853 
9.33  913 


9.33  974 

9.34  034 
9.34  095 
9.34  155 
9Ji4  215 
9.34  276 
9.34  336 
9.34  396 
9.34  456 
9.34  516 


9.34  576 
9.34  635 
9.34  695 
9.34  755 
9.34  814 


9.34  874 
9.34  933 

9.34  992 

9.35  051 
9.35  111 


9.35  170 
9.35  229 
9.35  288 
9.35  347 
9.35  405 
9.35  464" 
9.35  523 
9.35  581 
9.35  640 
9.35  698 


9.35  757 
9.35  815 
9.35  873 
9.35  931 
9.35  989 


L.  Cotg. 


L.  Cotg 


0. 67  253 
0.67  190 
0.67  128 
0.67  067 
0.67  005 


0.66  943 
0.66  881 
0.66  820 
0.66  758 
0.66  697 


0.  66  635 
0.66  574 
0.66  513 
0.66  452 
0.66  391 


0.66  330 
0.66  269 
0.66  208 
0.66  147 
0.  66  q87_ 
0.66  026 
0.65  966 
0.65  905 
0.65  845 
0.65  785 


L.  Cos. 


9.99  040 
9.  99  038 
9.99  035 
9.99  032 
9.99  030 


9.99  027 
9.99  024 
9.99  022 
9.99  019 
9.99  016 


9.99  013 
9.99  Oil 
9.99  008 
9.99  006 
9.99  002 


9.98  983 
9.98  980 
9.98  978 
9.98  975 


0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 


0.65  424 
0.65  365 
0.65  305 
0.65  245 
0.65  ISO 


0.65  126 
0.65  067 
0.65  008 
0.64  949 
0. 64  889 
0. 64  830 
0.64  771 
0.64  712 
0.64  653 
0.64  595 


0.64  536 
0.64  477 
0.64  419 
0.64  360 
0.64  302 


0.64  243 
0.64  185 
0.64  127 
0.64  069 
0.64  Oil 


0.63  953 
0.63  895 
0.63  837 
0.03  779 
0.63  721 
0.63.664 


9.98  953 
9.98  950 
9.98  947 


9.98  916 
9.98  913 
9.98  910 


C3 

62 

1,0 

1,0 

2,1 

2,0 

3,2 

3,1 

4,2 

4,1 

5,2 

5,2 

6,8 

6,2 

7,4 

7,2 

8,4 

8,3 

9,4 

9,3 

10,5 

10,3 

21,0 

20,7 

31,5 

31,0 

42,0 

41,3 

52,5 

51,7 

60 

59 

1,0 

1,0 

2,0 

2,0 

3,0 

8,0 

4,0 

3,9 

5,0 

4,9 

6,0 

5,9 

7,0 

6,9 

8,0 

7,9 

9,0 

8,8 

10,0 

9,8 

20,0 

19,7 

30,0 

29,5 

40,0 

39'3 

50,0 

49'2 

57 

56 

1,0 

0,9 

1,9 

1,9 

2,8 

2,8 

3,8 

3,7 

4'8 

4,7 

5,7 

5,6 

6,6 

6,5 

7,6 

7,5 

8,6 

8,4 

9,5 

9,3 

19,0 

18,7 

28,5 

28,0 

38,0 

37,3 

47,5 

46,7 

10,3 
31,0 
51,7 


3 

S  1 

59 

oS 

9,8 
29,5 
49,2 

9,7 
29,0 
48,3 

1,0 
2,0 
3,0 
4,1 
5,1 
6,1 
7,1 
8,1 
9,2 
10,2 
20,3 
30'5 
40,7 
50,8 


3,9 
4;8 
5,8 
6,8 
7,7 
8,7 
9,7 
19'3 
29'0 
38'7 
48'3 


0,9 
1,8 
2,8 
3,7 
4,6 
5,5 
6,4 
7,3 
8,2 
9,2 
18,3 
27,5 
36,7 
45,8 


10,0 
30,0 
50,0 


9,5 

28'5 
47'5 


77c 


LOGAEITHMS  OF  CIECULAR  FUIJGTIONS. 


267 


Table  XXXVI — Loganthmio  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gaa33'  Logaritlunio  and  Trigonometric  Tables.] 

13° 


9. 35  209 
9. 35  263 
9. 35  318 
9.  35  373 
9.  35  427 


9.  35  481 
9.35  536 
9.  35  590 
9.  35  644 
9.  35  698 


9.35  752 
9.35  806 
9. 35  860 
9.  35  914 
9.  35  968 


9.  36  022 
9.  36  075 
9.36  129 
9.36  182 
9.  36  236 


9.  36  289 
9. 36  342 
9. 36  395 
9. 36  449 


9. 37  081 
9.  37  133 
9,  37  185 
9.  37  237 
9. 37  289 


9. 37  341 
9. 37  393 
9. 37  445 
9.  37  497 
9. 37  549 


9. 37  600 
9.  37  652 
9.  37  703 
9. 37  755 
9.  37  806 


9.37  858 
9. 37  909 
9. 37  960 
9.  38  Oil 
9.  38  062 


L.  Tang. 


9.  36  336 
9. 36  394 
9. 36  453 
9,36  509 
9. 36  566 


9.  36  624 
9. 36  681 
9.  36  738 
9.  36  795 
9.36  852 


9.  36  909 
9.  36  966 
9.  37  023 
9. 37  080 
9. 37  137 


9. 37  193 
9.37  250 
9. 37  306 
9.  37  363 
9.  37  419 


9. 37  476 
9.  37  533 
9.  37  588 
9.  37  644 
9.  37  700 


9.  37  756 
9.37  813 
9. 37  868 
9. 37  924 


9.  38  918 
9.  38  972 
9. 39  027 
9.  39  082 


9.  39  136 
9.  39  190 
9. 39  245 
9.  39  299 
9. 39  353 


9. 39  407 
9. 39  461 
9.  39  515 
9.  39  569 
9.  39  623 
9. 39  677 


L.  Cotg. 


0.  63  664 
0.  63  606 
0.  63  548 
0.  63  491 
0.  33  434 


0. 


376 


0.  63  319 
0.  63  262 
0.63  205 
0.63  148 


0.63  091 
0. 63  034 
0.  62  977 
0.  02  920 
0.  62  863 


0.  62  807 
0.62  750 
0.  62  694 
0. 62  637 
0. 62  581 


0. 63  524 
0.  63  468 
0.  62  413 
0.  63  356 
0.  63  300 


0.63  244 
0.62  188 
0.63  133 
5.  62  076 
0.  62  020 


0.  61  965 
0.61  909 
0.  61  853 
0.  61  798 
0. 61  743 


0.  61  687 
0.  61  632 
0.  61  577 
0.  61  521 
0. 61  466 


0.61  411 
0.  61  356 
0. 61  301 
0. 61  246 
0.  61  192 


0.  61  137 
0.  61  082 
0.  61  028 
0.60  973 
0.  00  918 


0. 60  864 
0. 60  810 
0. 60  755 
0.  60  701 
0.  60  647 


0.60  593 
0.  60  539 
0. 60  485 
0.60  431 
0. 60  377 
0.60  323 


9.  98  858 
9.93  855 
9.  98  852 


9.  98  843 
9.  98  840 
9.  98  837 
9.  98  834 
9.  98  831 


9.  98  828 
9.  98  825 
9. 98  822 
9.  98  819 
9. 98  816 


9.  98  813 
9.  98  810 
9.  98  807 
9.  98  804 
9.98  801 


9.98  798 
9.  98  795 
9.  98  792 
9.  98  789 
9.  98  786 


9.  98  783 
9, 98  780 
9.  98  777 
9.  98  774 
9.  98  771 


9. 98  768 
9. 98  765 
9.  98  763 
9.  98  759 
9. 98  756 


9.98  753 
9.  98  750 
9.  98  746 
9. 98  743 
9.  98  740 


9.  98  737 
9.  98  734 
9.  98  731 
9.  98  728 
9.  98  725 


9.  98  722 
9. 98  719 
9. 98  715 
9.98  712 
9. 08  709 


9.  98  706 
9.  98  703 
9.  98  700 


S7 

36 

1,0 

0,9 

1,9 

1,9 

2,8 

2,8 

3,8 

3,7 

4,8 

4,7 

5,7 

5,6 

6,6 

6,5 

7,6 

7,5 

8,6 

8,4 

9,5 

9,3 

19,0 

18,7 

28,6 

28,0 

33,0 

37,3 

47,5 

46,7 

Si 

53 

0,9 

0,9 

1,8 

1,8 

2,7 

2,6 

3,6 

3,5 

4,5 

4,4 

5,4 

5,3 

6,3 

6,2 

7,2 

7,1 

,      8,1 

8,0 

9,0 

8,8 

18,0 

17,7 

27,0 

26,5 

36,0 

35,3 

45,0 

44,2 

51 

4 

3 

0,8 

0,1 

0,0 

1,7 

0,1 

0,1 

2,6 

0,2 

0,2 

3,4 

0,3 

0,2 

4,2 

0,3 

0,2 

5,1 

0,4 

0,3 

6,0 

0,5 

0,4 

6,8 

0,5 

0,4 

7,6 

0,6 

0,4 

8,5 

0,7 

0,5 

17,0 

1,3 

1,0  ' 

35,5 

2,0 

1,5 

34,0 

2,7 

2,0  1 

42,5 

3,3 

2,5  I 

3^ 
S6 

3 
53 

9,3 

28,0 
46,7 

9,2 
27,5 
45,8 

2,8 
3,7 
4,6 
5,5 
6,4 
7,3 
8,2 
9,2 
18,3 
27,5 
36,7 
45,8 


6,9 
7,8 
8,7 
17,3 
26,0 
34,7 
43,3 


4 

4 

3 

53 

34 

58 

6,9 

6,8 

9,7 

20,6 

20,2 

29,0 

34,4 

33,8 

48,3 

48,1 

47,2 

— 

9,0 
27,0 
45,0 


76= 


268 


A  MANUAL  OJ?  TOPOGEAPHIO  METHODS. 


Table  XXXVI. — Logarithmic  sines,  cosinen,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.) 


51 

53 

0,9 

0,9 

1,8 

1,8 

3,7 

2,6 

3,6 

3,5 

4,5 

4,4 

5,4 

5,3 

6,3 

8,2 

7,2 

7,1 

8,1 

8,0 

9,0 

8,8 

18,0 

17,7 

27,0 

20,5 

36,0 

35,3 

45,0 

44,2 

51 

50 

0,8 

0,8 

1,7 

1/7 

2,6 

2,5 

3,4 

3,3 

4,2 

4,2 

5,1 

5,0 

6,0 

5,8 

6,8 

6,7 

7,6 

7,5 

8,5 

8,3 

17,0 

16,7 

25,5 

25,0 

34,0 

33,3 

42,5 

41,7 

48 
0,8 

47 

0,8 

i 

0,1 

1,6 

1,6 

0,1 

2,4 

214 

0,2 

3,2 

3,1 

0,3 

4,0 

3,9 

0,3 

4,8 

4,V 

0,4 

5,6 

5,5 

0,5 

6,4 

6,3 

0,5 

7,2 

7,0 

0,6 

H,0 

7,8 

0,7 

16,0 

15,7 

1,3 

24,11 

i;:),5 

2,0 

32,U 

■■UrI 

'■^,7 

4U,ll 

:jo,2 

3,3 

3,5 
4,3 
5,2 
6,1 
6,9 
7,8 
8,7 
17,3 
26,0 
34,7 
43,3 


1,6 
2,4 
3,3 
4/1 
4,9 
5,7 
6,5 
7,4 
8,2 
16,3 
24,5 
32,7 
40,8 

3 

0,0 
0,1 
0,2 
0,2 
0,2 
0,3 
0,4 
0,4 
0,4 
0,5 
1,0 
1,5 
2,0 
2,5 


4 

4 

4 

1     ^* 

.53 

52 

6,8 

6,6 

6,5 

20,2 

19,9 

19,5 

33,8 

33,1 

32,5 

!   47,2 

46,4 

45,5 

3 

3 

3 

54 

53 

52 

i      9,0 

!    27,0 

45,0 

8,8 
26,5 
44,2 

8,7 
26,0 
43,3 

6,4 
19,1 
31,9 
44,6 


8,5 
25,5 
42,5 


75= 


LOGARITHMS  OF  OlECULAK  ru:NGTIO:tfS. 


269 


Table  XXXVI. — LogarUhniie  sines,  cosines,  tangentSj  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

15° 


9. 41  300 
9.  41  347 
9.41  394 
9.41  441 
9.41  488 


9.41  635 
9.41  582 
9.41  628 
9.41  675 
9.41  722 


9.41  768 
9.41  815 
9.41  861 
9.41  908 
9.41  954 


9.42  001 
9.42  047 
9.42  093 
9.42  140 
9.42  186 


9.42  232 
9.42  278 
9.42  324 
9.42  370 
9.  42  416 


9. 42  461 
9.42  507 
9.42  553 
9.42  599 
9.42  644 


9.42  690 
9. 42  735 
9. 42  781 
9.  42  826 
9. 42  872 


9. 42  917 

9.42  962 

9. 43  008 
9.43  053 
9. 43  098 


9.43  143 
9.43  188 
9.  43  233 
9.43  278 
9.43  323 


9.43  367 
9.43  412 
9.43  457 
9.43  502 
9.43  546 


9.43  591 
9.43  635 
9.43  680 
9.43  724 
9.43  769 


9.43  813 
9.43  857 
9.43  901 
9.43  946 

9.43  990 

9. 44  034 


9.43  U57 
9. 43  108 
9.43  1.58 
9.43  208 
9.43  258 
9.43  308 
9.  43  358 
9. 43  408 
9.43  458 
9.43  508 


9.43  658 
9.43  607 
9.43  657 
9.43  707 
9.43  756 


9.44  787 
9. 44  836 
9. 44  884 
9.  44  933 
9. 44  981 


9.  45  029 
9.45  078 
9.45  126 
9.  45  174 
9.45  222 


9.45  271 
9.45  319 
9.45  367 
9.45  415 
9.45  463 


9.  45  511 
9.45  559 
9.45  gu6 
9.45  654 
9.45  702 
9.45  IW 


L.  Cotg. 


0.  57  195 
0. 57  144 
0. 57  094 
0.  57  043 
0.56  993 


0.  56  943 
0. 56  892 
0.  56  842 
0.  56  792 
0. 56  742 


0.  56  692 
0.  56  042 
0.  56  592 
0.50  542 
0. 56  492 


0.  .56  442 
0. 56  393 
0.  56  343 
0.  56  293 
0.  56  244 
0.  56  194 
0.56  145 
0.56  095 
0.  56  040 


0.55  947 
0.  55  898 
0.55  849 
0.  .55  799 
0^5^750 
6. 55  701 
0. 55  652 


0.; 


603 


0.  55  456 
0.  55  408 
0.  55  359 
0. 55  310 


0.  55  164 
0.55  116 
0. 55  067 
0.  55  019 


0. 54  971 
0.  54  922 
0.  54  874 
0. 54  826 
0.  54  778 


0. 54  729 
0.54  681 
0. 54  633 
0.  54  585 
0. 54  537 


0. 54  489 
0. 54  441 
0.54  394 
0.54  346 
0. 54  298 
0.  54  250 


9.  98  494 
9.  9,S  491 
9.98  488 
9. 98  484 
9.  98  481 


9.  98  477 
9.  98  474 
9.  98  471 
9.98  467 
9.  98  464 


9.  98  373 
9.  98  370 
9. 98  366 
9.  f  8  363 
9. 98  359 


9.  98  356 
9.  98  352 
9.  98  349 
9. 98  345 
9. 98  342 


9.98  338 
9. 98  334 
9.98  331 
9.  98  327 
9.  98  324 


9.  98  320 
9.98  317 
9.  98  313 
9.  98  309 
9. 98  306 


9.98  302 
9.  98  299 
9.98  295 
9.98  291 
9.98  288  I 


L.  Tang.    L.  Sin.  I  d. 


740 


51 

50 

0,8 

0,8 

1,7 

1,7 

2,6 

2,5 

3,4 

3,3 

4,2 

4,2 

5,1 

5,0 

6,0 

5,8 

6,8 

6,7 

7,6 

7,5 

8,5 

8,3 

17,0 

16,7 

25,5 

25,0 

34,0 

-    33,3 

42,5 

41,7 

48 
0,8 

47 

0,8 

1,6 

1,6 

2,4 

2,4 

3,2 

3,1 

4,0 

3,9 

4,8 

4,7 

5,6 

5,5 

6,4 

6,3 

7,2 

7,0 

8,0 

7,8 

16,0 

15,7 

24,0 

23,5 

32j0 

31,3 

40,0 

39,2 

45 

0,8 

44 

0,7 

4 

0,1 

1,5 

1,5 

0,1 

2,2 

2'2 

0,2 

3,0 

2,9 

0,3 

3,8 

3,7 

0,3 

4,5 

4,4 

0,4 

5,2 

5,1 

0,5 

6,0 

5,9 

0,5 

6,8 

6,6 

0,6 

7,5 

7,3 

0,7 

15,0 

14,7 

1,3 

22,5 

22,0 

2,0 

30,0 

29,3 

2,7 

37,5 

36,7 

3,3 

16,3 
24,5 
32,7 
40,8 


7,7 
15,3 
23,0 
30,7 
38,3 

3 

0,0 
0,1 
0,2 
0,2 
0,2 
0,3 
0,4 


4 

1 

4 

50 

49 

48 

6,2 
18,8 
31,2 
43;8 

6,1 
18,4 
30,6 
42,9 

6,0 
18,0 
30,0 
42,0 

51 

50 

49 

8,5 

1    25,5 

42,5 

8,3 
25,0 
41,7 

8,2 
24,5 
40,8 

5,9 
17,0 
29,4 
41,1 


8,0 
24,0 
40,0 


270 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents — C'ontiuueil. 
IBxtracted  from  Gauss'  Logaritlmiic  ami  Trigonometric  Tables.] 

16° 


48 

47 

0,8 

0,8 

1,6 

1,6 

2,4 

2,4 

3,2 

3,1 

4,0 

3,9 

4,8 

4,7 

5,6 

5,5 

6,4 

'6,3 

7,2 

7,0 

8,0 

7,8 

16,0 

15,7 

24,0 

23,5 

32,0 

31,3 

40,0 

39,2 

45 

44 

0,8 

0,7 

1,5 

1,5 

2,2 

2,2 

3,0 

2,9 

3,8 

3,7 

4,5 

4,4 

5,2 

5,1 

6,0 

5,9 

6,8 

6,6 

7,5 

7,3 

15,0 

14,7 

22,5 

22,0 

30,0 

29,3 

37,5 

36,7 

42 

41 

4 

0,7 

0,7 

0,1 

1,4 

1,4 

o,u 

2,1 

2,0 

0,2 

2,8 

2,7 

0,3 

3,5 

3,4 

0,3 

4,2 

4,1 

0,4 

4,9 

4,8 

0,5 

5,6 

5,5 

0,5 

6,3 

0,2 

0,6 

7,0 

6,8 

0,7 

14,0 

13,7 

1,3 

21,0 

20,5 

2,0 

28,0 

27,3 

2,7 

35,0 

34,2 

0,8 
1,5 
2,3 
3,1 
3,8 
4,6 
5,4 
6,1 
6,9 
7,7 
15,3 
23,0 
30,7 


0,7 
1,4 
2,2 
2,9 
3,6 
4,3 
5,0 
5,7 
6,4 
7,2 
14,3 
21,5 
28,7 
35,8 


4 

4 

48 

47 

6,0 
18,0 
30,0 
42,0 

5,9 
17,6 
29,4 
41,1 

.•5 

48 

3 

47 

8,0 
24,0 
40,0 

7,8 
23,5 
39,2 

5,6 
16,9 
28,1 
39,4 


7,7  7,5 
23,0  22,5 
38,3      37,5 


73° 


LOGAEITHMS  OF  CIECULAR  FUNCTIONS. 


271 


Table  liXXYl,~~Loganthmic  sinesy  cosineSj  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometiic  Tablea.] 

17° 


9. 46  670 
9.  46  717 
9.46  758 
9.46  800 
9.  46  841 
9. 46  882 
9.  46  923 
9. 46  964 


9.47  045 
9.  47  086 
9.  47  127 
9.47  168 


9.  47  209 
9.47  249 
9.  47  290 
9. 47  330 
9. 47  371 


9.  47  411 
9. 47  432 
9. 47  492 
9.47  533 
_9^47_57£ 
9.  47  6i3 
9.  47  654 
9.47  694 


9.47  814 
9.  47  854 
9.  47  894 
9.  47  934 
9^  47  974 

9.48  014 
9.  48  054 
9.  48  094 
9.48  133 
9. 48  173 


9.48  213 
9.48  252 
9. 48  292 
9.48  332 
9.  48  371 


9.  48  411 
9.  43  450 
9. 48  490 
9.48  529 


9.48  803 
9.  48  842 
9. 48  881 
9.  48  920 
9. 48  959 


1.48  534 
1.48  579 
i.  48  624 
i.  48  669 
I.  48  714 
I.  48  759 
I.  48  804 
:.  48  849 
1.48  894 
1.48  939 


1.48  984 

1.49  029 
I.  49  073 
1.49  118 
1.49  163 


'.  49  207 
I.  49  252 
'.49  296 
.49  341 
'.  49  385 


'.  49  430 
1. 49  474 
'.  49  519 
.49  563 
.  49  607 


9.  49  652 
9.  49  696 
9.49  740 
9.  49  784 
9.  49  828 


i.  50  180 
I.  50  223 
1. 50  267 
'.  50  311 
.  50  355 
.  50  398 
.  50  442 
.  50  485 


I.  50  529 
1.50  572 
1.  50  616 
I.  50  659 
I.  50  703 
I.  50  746 
I.  50  789 
I.  50  833 
I.  50  876 
I.  50  919 

1. 50  962 

1.51  005 
'.  51  048 
I.  51  092 
.51  135 


0.51  466 
0.51  421 
0. 51  376 
0.  51  331 
0.  51  286 


9, 98  056 
9. 98  052 
9. 98  048 
9. 98  044 


0.51  241 
0. 51  196 
0.51  151 
0.  51  106 
0.  51  061 


0. 51  016 
0.50  971 
0. 50  927 
0. 50  882 
_0.J0_837 
0. 50  793 
0. 50  748 
0.  50  704 
0.  50  659 
0.50  615 


9.  98  036 
9.98  032 
9.98  029 
9.98  025^ 
9. 98  021 
9. 98  017 
9.  98  013  : 
9. 98  009 
9. 98  005 


0.50  570 
0.  50  526 
0. 50  481 
0. 50  437 
0.  50  393 


0.  50  348 
0.  50  304 
0.  50  260 
0. 50  216 
0.  50  172 


0.50  128 
0.  50  084 
0.50  040 
0.49  996 
0.49  952 


0.49  908 
0.49  804 
0.49  820 
0.49  777 
0. 49  733 


0.49  689 
0.49  645 
0.  49  602 
0.  49  558 
0.49  515 


0.  49  471 
0.  49  428 
0.49  384 
0.  49  341 
0.  49  297 


0. 49  254 
0.49  211 
0.49  167 
0.49  124 
0. 49  081 
6.49  038 
0. 48  995 
0.48  952 
0. 48  908 
0. 48  865 
0.  48  822 


9.98  001  I 
9.  97  997  1 
9.  97  993  i 
9. 97  989 
9.  97  986 


9. 97  982 
9.97  978 
9. 97  974  ■ 


9.97 


70 


9.  97  966_ 
9. 97  962 
9.  97  958 
9.  97  954 
9. 37  950 
9.97  946 


9.97  942 
9. 97  938 
9.97  934  ! 
9.  97  930 
9. 97  926  ' 


9. 97  910 
9. 97  906 


9.  97  902 
9.  97  898 
9. 97  894 
9.  97  890  I 
9. 97  886 


9. 97  861 
9.  97  857 
9.  97  8.53 
9.  97  849 
9.  97  845 


9.  97  841 
9. 97  837 
9.  97  833 
9. 97  829 
9.  97  825 


L.  Tang.  I  L.  Sin. 


45 

44 

0,8 

0,7 

1,5 

1,5 

2,2 

2,2 

3,0 

2,9 

3,8 

3,7 

4,5 

4,4 

5,2 

5,1 

5,9 

6,8 

6,6 

7,5 

7,8 

15,0 

14,7 

22,5 

22,0 

30,0 

29,3 

37,5 

36,7 

42 

41 

0,7 

0,7 

1,4 

1,4 

2,1 

2,0 

2,8 

2,7 

3,5 

3,4 

4,2 

4,1 

4,9 

4,8 

5,6 

5,5 

6,3 

6,2 

7,0 

6,8 

14,0 

13,7 

21,0 

20,5 

28,0 

27,3 

3b,0 

34,2 

39 

5 

4 

0,6 

0,1 

0,1 

1,3 

0,2 

0,1 

2,0 

0,2 

0,2 

2,6 

0,3 

0,3 

3,2 

0,4 

0,3 

3,9 

0,5 

0,4 

4,6 

0,6 

0,5 

5,2 

0,7 

0,5 

5,8 

0,8 

0,6 

6,5 

0,8 

0,7 

13,0 

1,7 

1,3 

19,5 

2,5 

2,0 

26,0 

3,3 

2,7  ! 

32,5 

4,2 

3,3 

5 

4 

43 

45 

4,3 

5,6 

12,9 

16,9 

21,5 

28,1 

30,1 

39,4 

38,7 

- 

0,7 
1,4 
2,2 
2,9 
3,6 
4'3 
5,0 
5,7 
6,4 
7,2 
14,3 
21,5 
28,7 
35,8 


4,0 
4,7 
5,3 
6,0 
6,7 
13,3 
20,0 
26,7 
33,3 


0,0  ! 

0,1  I 

0,2 

0,2 

0,2 

0,3 

0,4 

0,4 

0,4 

0,5 

1,0 

1,5 

2,0 

2,5 


5,5 
16,5 
27,5 


4 

3 

3 

4S 

45 

44 

5,4 

7,5 

7,3 

16,1 

22,5 

26,9 

37,5 

36,7 

37,6 



■ 

P 

P. 

72^ 


272 


A  MANUAL  OF  TOrOGKAPHIO  METHODS. 


Table  XXXVl.^ Lof/anthmic  !<'ntes,  cosines,  lanfjentH,  and  cotangents — Contiuued, 
[Extraotod  Iroui  Gauss"  Loi^iirithmic  ami  Trigonometric  Tables.] 

18° 


9.48  998 

9.49  037 
9. 49  07(i 
9.49  115 
9.49  153 

9. 49  192 
9.49  231 
9.49  269 
9.49  308 
9.49  347 


9.  49  385 
9. 49  424 
9. 49  462 
9.  49  500 
9.49  539 


9.49  577 
9.49  615 
9.49  654 
9.49  692 
9.49  7J0 


9.  49  76S 
9. 49  806 
9.49  844 
9. 49  882 
9.  49  920 


9.49  958 

9.49  996 

9.50  034 
9. 50  072 
9.50  110 


9.  50  148 
9.  50  185 
9.  50  223 
9.50  261 
9. 50  298 


9.  50  336 
9.  .50  374 
9.50  411 
9.  50  449 
9.50  480 


9. 50  523 
9.50  561 
9.  50  598 
9. 50  635 
9.  50  673 


9.50  710 
9.50  747 
9.50  784 
9.50  821 
9.50  8.58 


9. 50  890 
9.  50  933 

9.50  970 

9.51  007 
9.  51  043 


9.51  080 
9.51  117 
9.51  154 
9.51  191 
9.51  227 
9.  51  264 


L.  Tang. 


9.  51  606 
9,51  648 
9.51  691 
9.51  734 
9.  51  776 


9.51  819 
9.51  861 
9.51  903 
9.51  946 
9.51  988 


9.  52  031 
9.  52  073 
9.52  115 
9.52  157 
9.^2  200 
9.52  242 
9. 52  284 
9.52  326 
9.  52  368 
9^52  410 
9.  52  452 
9.  52  494 
9. 52  536 
9.52  578 
9.  52  620 


9.  52  661 
9.  52  703 
9.  52  745 
9.52  787 
9.52  829 


9.53  078 
9.53  120 
9.53  161 
9.53  202 
9.53  244 


9.53  492 
9.53  533 
9.53  574 
9.53  615 
9.  53  656 
9.  53  697 


L.  Cotg. 


L.  Cotg. 


0.  48  822 
0,  48  779 
0.  48  736 
0.48  694 
0.  48  651 


0.48  008 
0.48  505 
0.48  ,522 
0.48  480 
0.48  437 


0. 48  394 
0.  48  352 
0.48  309 
0.  48  266 
0.48  224 


0.48  181 
0.48  139 
0.48  097 
0.48  0.54 
0.48  012 


0.47  069 
0.47  927 
U.47  885 
0.47  343 
0.47  800 


0.  47  716 
0.47  674 
0.  47  632 
0.  47  590 


0.  47  548 
0. 47  506 
0.47  464 
0.47  422 
0.47  380 


0.47  339 
0.47  297 
0.47  255 
0.47  213 
0.47  171 


0.  47  130  j 
0,  47  088 
0. 47  047 
0.  47  005 
0.  46  963 


0. 46  922 

0.  46  880 

0.  46  839 

0.  46  798 

0.  46  7.56 


0.46  715 
0.  46  673 
0.  46  632 
0.  46  591 
0. 46  550  ! 


9.  97  821 
9.97  817 
9.97  812 


9.97  767 
9.97  763 


9.97  759 
9.97  7.54 
9.97  750 
9.  97  746 
9.97  742 


9.  97  738 
9.97  734 
9.97  729 
9.97  725 
9^7  731 
9.  97  717 
9.  97  713 
9.97  708 
9.  97  704 
9.97  700 


9. 97  696 
9.  97  691 
9. 97  687 
9.97  683 
9.97  679 


9.97  674 
9.97  670 
9.97  666 
9.97  662 
9.  97  657 


9.97  653 
9.97  649 
9.97  645 
9.97  640 
9.97  636 


9.97  632 
9.97  628 
9.97  623 
9.97  619 
9.97  615 


9.97  610 
9.  97  606 
9  97  602 
9.97  .597 
9.97  593 


1.97  589 


0.46  508  I 
0.  46  467 
0.  46  426 
0.  46  385 
0.  46  344  I 
0.  46  303  i 


L.  Tang.  I  L.  Sin. 


43 

0,7 

42 

0,7 

1,4 

1,4 

2,2 

2,1 

2,9 

2,8 

3,6 

3,5 

4,3 

4,2 

5,0 

4,9 

.  5,7 

5,6 

6,4 

6,3 

7,2 

7,0 

14,3 

14,0 

21,5 

21,0 

28,7 

28,0 

35,8 

35,0 

.S9 

88 

0,6 

0,6 

1,3 

1,3 

2,0 

1,9 

2,6 

2,5 

3,2 

3,2 

3,9 

3,8 

4,0 

4,4 

5,2 

5,1 

5,8 

5,7 

6,5 

6,3 

13,0 

12,7 

19,5 

19,0 

26,0 

25,3 

32,5 

31,7 

30 

5 

0,6 

0,1 

1,2 

0,2 

1,8 

0,2 

2,4 

0,3 

3,0 

0!4 

3,6 

0,5 

4,2 

0,6 

4,8 

.0,7 

5,4 

0,8 

6,0 

0,8 

12,0 

1,7 

18,0 

2,5 

24,0 

3,3 

30,0 

4,2 

5 

.5 

43 

43 

4,3 ; 

4,2 

12,9  ; 

12,0 

21,5  1 

21,0 

30,1 

29,4 

38,7 

37,8 

0,7 
1,4 
2,0 
2,7 
3,4 
4,1 
4,8 
5,5 
6,2 
6,8 
13,7 
20,5 
27,3 
34,2 


4,1 
12,3 
20,5 
28,7 
36,9 


5,4 

5,2 

5,1 

16,1 

15,8 

15,4 

26'9 

26,2 

25,6 

37,6 

36,8  ; 

35,9 

71^ 


LOGAEITHMS  OF  CIECULAR  FUifCTIONS. 


273 


Table  XXXVI. — Logarithmic  sines,  cosines,  iangentSj  and  cotangents — Contiaiued. 
[Extracted  from  Gauss'  Logaritbmic  and  Trigonometric  Tables.] 

19° 


9.  51  264 
9.51  301 
9.51  338 
9.51  374 
9.51  411 


9.51  447 
9.51  484 
9.51  520 
9.51  557 
9.  51  593 
9.'51  629 
9.51  666 
9.  51  702 
9.51  738 
9.51  774 
9.51  811 
9.51  847 
9.51  883 
9.51  919 
9.51  955 


9. 52  527 
9.  52  563 
9.  52  598 
9.  52  634 
9.  52  669 


9.52  705 
9.  52  740 
9.52  775 
9.  52  811 
9.  52  846 


9.  58  021 


9. 53  056 
9.53  092 
9.53  126 
9.53  161 
9.53  196 


9.  53  231 
9.53  266 
9.  53  301 
9.53  336 
9. 53  370 
9. 53  405 


d.  c.  !  L.  Cotg. 


9. 53  697 
9.  53  738 
9.53  779 
9.53  820 
9.53  861 
9.  53  902 
9.53  943 
9.  53  981 
9.  54  025 
9.  51  0C5 


9.54  106 
9.54  147 
9.  .54  187 
9.  54  228 
9.54  269 


9.54  309 
9.  .54  350 
9.54  390 
9.54  431 
9.  54  471 


9.54  915 
9.51  955 
9.54  995 
9.  55  035 
9.  55  075 


9.55  115 
9.55  155 
9.55  195 
9. 55  235 
9. 55  275 


9.55  315 
9.55  355 
9.55  395 
9.  55  434 
9.-55  474 
9. 55  514 
9.  55  554 
9.55  593 
9.  55  633 
9.55  673 


9.  55  712 
9.55  752 
9.55  791 
9.55  831 
9.55  870 


L.  Cotg. 


0.46  303 
0.46  262 
0.46  221 
0.46  180 
0.46  139 


0.46  098 
0.  46  057 
0.46  016 
0.45  975 
0.  45  935 


0.45  894 
0.45  853 
0.45  813 


0.45  010 
0.  45  569 
0.45  .529 


0.45  488 
0.45  448 
0. 45  407 
0.45  367 
0.45  327 


0.45  286 
0.45  246 
0,45  206 
0.  45  165 
0.45  125 


0.45  085 
0.45  045 
0.45  005 
0.44  965 
0.44  925 


0.44  885 
0.44  845 
0.  44  805 
0.44  765 
0.44  725 


0.44  685 
0.44  645 
0.44  6115 
0.44  566 
0.  44  526 


0.  44  486 
0.  44  446 
0,44  407 
0. 44  367 
0.44  327 


0.44  288 
0.44  248 
0.  44  209 
0.44  169 
0.44  130 


0.44  090 
0,44  051 
0.44  Oil 
0.43  972 
0.43  933 
0. 43  893 


d.  c.   L.  Tang. 


9.97  435 
9.97  430 
9.97  426 
9.  97  421 
9.  97  417 


9.  97  412 
9.  97  408 
9.  97  403 
9.  97  399 
9.97  394 


9.  97  390 
9.97  385 
9.97  .381 
9.97  376 
9.  97  372 
9.97  367 
9.97  363 
9.  97  358 
9.97  353 
9.97  349 


9.97  340 
9.97  335 
9.97  331 
9.97  326 


9.97  322 
9.97  317 
9.97  312 
9.97  308 
9.97  303 
9. 97  299 


41 

40 

0,7 

0,7 

1,4 

1,3 

2,0 

2,0 

2,7 

2,7 

3,4 

3,3 

4,1 

4,0 

4,8 

4,7 

5,5 

5,3 

6,2 

6,0 

6,8 

6,7 

13,7 

13,3 

20,5 

20,0 

27,3 

26,7 

34,2 

33,3 

37 

36 

0,6 

0,6 

1,3 

1,2 

1,8 

1,8 

2,5 

2,4 

3,1 

3,0 

3,7 

3,6 

4,3 

4,2 

4,9 

4,8 

5'6 

5,4 

6,2 

6,0 

12,3 

12,0 

18,5 

18,0 

24'? 

24,0 

30,8 

30,0 

34 

5 

0,6 

0,1 

1,1 

0,2 

1,7 

0,2 

2,3 

0,3 

2,8 

0,4 

3,4 

0,5 

4,0 

0,6 

4,5 

0,7 

5,1 

0,8 

5,7 

0,8 

11,3 

1,7 

17,0 

2,5 

22,7 

28,3 

4,2 

5 

5 

41 

40 

4,1 

4,0 

12,3 

12,0 

20,5 

20,0 

28,7 

28,0 

36,9 

36,0 

41 

40 

5,1 

5,0 

15,4 

15,0 

25,6 

25,0 

35,9 

35,0 

6,5 
13,0 
19,5 
26,0 
32,5 


5,2 
5,8 
11,7 
17,5 
23'3 
29,2 


11,7 
19,5 
27,3 
35,1 


4,9 
14,6 
24,4 
34,1 


MON   XXII- 


-18 


70= 


274 


A  MAXUAL  OF  TOPOGRAPHIC  METHODS. 


Taiu.k  WWl.—Lundrithmic  shies,  roslnes,  taHi/euts,  mid  mlaiii/cnts—Contiuned. 
tExtractfil  from  lliiuss'  Logarithmic  iim\  Trigouomctric  Tables.] 

20° 


L.  Sin. 

rt. 

L.  Tang. 

(I.C. 

L.  Cotg. 

L.  Cos. 

a. 

P. 

P. 

0 

9.  53  405 

9.  56  107 

0.  43  893 

9.  97  299 

5 
5 
4 
5 
4 
5 
5 
4 

60 

40 

39 

0,6 
1,3 

2,e 

38 
0,6 

I 

9.53  440 

35 

.  9.  56  146 

0.  43  854 

9.  97  294 

59 

1,3 
2,0 
•2,7 

1,3 
1,9 
2,5 
3,2 
3,8 

9.53  475 

35 

9.  56  185 

0.43  815 

9. 97  289 

58 

3 

9. 53  509 

9.56  224 

0.43  776 

9. 97  285 

2,6. 
3,2 
3,9 

4 

9.  53  544 
9. 53  578 

34 

9. 56  264 

39 

0.  43  736 

9.97  276 

5 
6 

3,3 

4,0 

5 

9.  56  303 

0.  43  697 

55 

6 

9.  53  613 

35 

9.56  342 

0.43  658 

9.  97  271 

54 

7 

4,7 

4,6 

4,4 

9.53  647 

34 

9.  50  381 

0.  43  619 

9.  97  266 

53 

8 

5,3 

5,2 

.  5,1 

S 

9.53  682 

35 

9.56  420 

0.43  580 

9.  97  262 

52 

9 

6,0 

5,8 

5,7 

9 

9,  53  716 

34 
35 

9.  56  459 
9.56  498 

39 

0.43  541 

5 

10 
20 

6,7 
13,3 

6,5 
13,0 

6,3 
12,7 

10 

e.  43  502 

9.  97  252 

50 

11 

9  53  785 

34 

9.  56  537 

0.-43  463 

9.  97  248 

49 

30 

20,0 

19,5 

19,0 

12 

9.  53  819 

34 

9.56  576 

0.43  424 

9.  97  243 

5 
4 

48 

40 

26,7 

13 

9.53  854 

35 

9. 50  615 

0.43  385 

9.  97  238 

47 

50 

14 

9.53  888 
0.  53  922 

34 
34 

9.  56  654 

39 

0. 43  346 

9.97  234 

46 

1 

37 

0,6 

35 

0,6 

34 

0,0 

15 

9.56  693 

0.  43  307 

9. 97  229 

45 

16 

9.53  957 

35 

9.  5B  732 

0.43  268 

9.97  224 

4 
5 
5 
4 

44 

2 

1,2 

1,2 

1,1 

17 

9.53  991 

34 

9.  56  771 

0.  43  229 

9. 97  220 

43 

3 

1,8 

1,8 

1,7 

18 

9.  54  025 

34 

9.  56  ,S10 

0.43  190 

9.97  215 

42 

4 

2,5 

2,3 

2,3 

19 

9. 54  059 

34 
34 

9.  56  849 

38 

0.43  151 

■9.97  210 
9. 97  206 

41 

5 
6 

8,1 

3,7 

2,9 
3,5 

2,8 
3,4 

20 

9.56  887 

0.43  113 

40 

9.54  127 

34 

9.56  926 

39 

0. 43  074 

9. 97  201 

5 
4 
5 

39 

7 

f'? 

*,1 

4,0 

22 

9.54  161 

34 

9.  .56  965 

39 

0.43  035 

9. 97  196 

38 

8 

4,9 

4,7 

23 

9.54  195 

34 

9. 57  004 

0. 42  996 

9.97  192 

37 

24 

9.  54  229 

34 

9.57  042 

39 
39 
38 
39 

0. 42  958 

9.97  187 

36 

20 
30 
40 
50 

12,3 
18,5 
24,7 
30,8 

11,7 
17,5 
23,3  . 
29,2 

11,3 
17,0 
22,7 
28,3 

26 
28 

9.  .54  203 
9. 54  297 
9.  54  331 
9.  54  365 

34 
34 
34 
34 

9.  57  081 
9.  57  120 
9.57  158 
9.57  197 

0.42  919 
0.42  880 
0.  42  842 
0. 42  803 

9.97  182 
9.97  178 
9.  97  173 
9.97  168 

4 
5 
5 

35 
34 
33 
32 

29 

9. 54  399 

34 
34 

9.  57  235 
9.57  274 

39 

0.42  765 
0.42  726 

9.97  163 
9.  97"  1.59 

4 

1 

33 

0,6 

5 

0,1 

i 

0,1 

30 

9.54  433 

SO 

31 

9.54  466 

33 

9.  57  312 

0.  42  688 

9.  97  154 

^ 

29 

32 

34 

9.57  351 

0. 42  649 

9.97  149 

4 

5 
5 

28 

33 

9.54  534 

34 

9.57  389 

0.  42  611 

9.  97  145 

27 

34 

9.  54  567 

33 
34 

9.57  428 

38 

0.  42  572 

9.97  140 

26 

5 
6 
7 
8 
9 
10 
20 
30 

2,8 
3,3 
3,8 
4,4 
5,0 
5,5 
11,0 
16,5 

0,4 
0,5 
0,6 
0,7 
0,8 
0,8 
1,7 
2,5 

0,4 
0,5 
0,5 
0,6 
0,7 
1,3 
2,0 

35 

9.54  6U1 

9. 57  466 

0.42  534 

9.97  135 

25 

36 

9.54  635 

34 

9.  57  504 

0.42  496 

9.97  130 

4 

37 

0.  54  668 

33 

9.  57  543 

0.  42  457 

9.97  126 

38 

9.54  702 

34 

9.  57  581 

0.42  419 

9.97  121 

^ 

22 

39 

9.  54  735 

33 
34 

9.57  619 

39 

0.42  381 

9.97  116 

5 

40 

9. 54  769 

9.57  658 

0.  42  342 

9.97  111 

ao 

41 

9.  54  802 

33 

9.57  696 

0.  42  304 

40 

22,0 

3,3 

2,7 

42 

9.54  836 

34 

9.  57  734 

0. 42  266 

9.97  102 

18 

50 

27,5 

4,2 

3,3 

33 

9.57  772 

0.42  228 

5 
5 
4 
5 
5 

44 

9.54  903 

34 
33 

9.  .57  810 

38 
39 

0.42  190 

9.97  092 

16 

5 

5 

^ 

45 

9.54  930 

9.57  849 

0.42  151 

9.97  087 

15 

46 

9.54  969 

33 

9.57  887 

0.42  113 

9.97  083 

14 

47 

9.  55  003 

34 

9.57  925 

0.42  075 

9.  97  078 

13 

3,9 

3,8 

48 

9.55  036 

33 

9.  57  963 

0.42  037 

9.  97  073 

12 

12,0 

11,7 

11,4 

49 

9.55  069 

33 
33 

9.  58  001 
9. 58  039 

38 

0.41  999 

5 
4 
5 
5 

3 
4 
5 

•20,0 
28,0 

19,5 
27,3 

19,0 
26,6 

50 

9.55  102 

0.41  961 

9.  97  063 

10 

51 

9.55  136 

34 

9.  58  077 

0.41  923 

9.97  059 

36,0 

35,1 

34,2 

9. 55  169 

33 

9.58  115 

38 

0.4L  885 

9. 97  054 

8 

53 

9.55  202 

33 

9.58  153 

0.41  847 

9.97  049 

7 

5 

4 

4 

54 

9. 55  235 

33 
33 

9.58  191 

38 

0.41  809 

9. 97  044 

5 
4 

6 

37. 

39 

3S 

55 

9. 55  263 

9.  58  229 

0.41  771 

9. 97  039 

5 

56 

9. 55  301 

33 

9.  58  267 

38 

0.41  733 

9.97  035 

4 

0 

3,7 

4,9 

4,8 

57 

9.55  334 

33 

9.58  304 

0.41  696 

9.97  030 

3 

11,1 

14,6 

14,2 

58 

9.55  367 

33 

9.  58  342 

0.41  658 

18,5 

24,4 

23,8 

59 

9. 55  400 

33 
33 

9.58  380 
9. 58  418 

38 

0.41  620 

5 

4 
5 

25,9 
33,3 

34,1 

33,2 

60 

9. 55  433 

0.41  582 

9.  97  015 

0 

1.  Cos. 

(1. 

L.  Cotg. 

a.c. 

L.  Tang. 

L.  Sin. 

d. 

' 

P 

.P. 

69<^ 


LOGARITHMS  OF  CIEOITLAR  FUNCTIONS. 


275 


Table  XXXVI. — Lorjarithmio  sines,  cosines,  tangents,  and  cotanrjents- 
[Estracted  from  Gauss'  Logaritlimic  .aud  Trigonometric  Tables.] 

21° 


L.  Sin. 

d. 

L.  Tang. 

d.  c. 

L.  Cotg. 

L,  Cos. 

0 

9.55  433 

9.  58  418 

0.  41  582 

9. 97  015 

1 

9.  55  466 

9.  58  455 

0.41  545 

9. 97  010 

9.  55  499 

9. 58  493 

0.  41  507 

9. 97  005 

3 

9.  55  532 

9.  58  531 

0.  41  469 

9. 97  001 

4 

9.  .55  564 

33 

9.  58  569 

37 

0.41  431 

•9.  96  996 

5 

9.55  597 

9.58  606 

0.41  394 

9. 96  991 

6 

9. 55  630 

9.  .58  644 

0.41  356 

9. 90  980 

7 

9.55  663 

9.  58  081 

0.  41  319 

9.  90  981 

8 

9.55  695 

9.58  719 

0.41  281 

9.  90  976 

9 

9. 55  728 

33 

9.  58  757 

37 

0.  41  243 

9.  96  971 

10 

9.55  761 

9.  58  794 

•  0.  41  206 

9.  96  966 

11 

9.55  793 

9.  58  832 

0.41  168 

9. 96  962 

12 

9.  55  826 

9.  58  809 

0.41  131 

9.  96  957 

13 

9.55  858 

9.  58  907 

0.41  093 

9. 96  952 

14 

9.55  891 

32 

9.  58  944 

37 

0.41  056 

9. 90  947 

15 

9. 55  923 

9.  58  981 

0.41  019 

9.  96  942 

16 

9. 55  956 

9. 59  019 

0.40  981 

9.  96  937 

17 

9.  55  988 

9.  59  056 

0.  40  944 

9. 96  932 

18 

■  9.56  021 

9. 59  094 

0.40  906 

9.  96  927 

19 
20 

9.  56  053 

32 

9.59  131 

37 

0.40  869 

9.  96  922 

9.56  085 

9.  .59  168 

0.  40  832 

9.  90  917 

21 

9.56  118 

9.59  205 

0.  40  795 

9, 90  912 

22 

9.56  1.50 

9.  .59  243 

0.40  757 

9. 90  907 

23 

9.  56  182 

9.50  280 

0. 40  720 

9.  96  903 

24 

9.56  215 
9.56  247 

32 

9. 59  317 

37 

0.40  683 

9.  96  898 

25 

9. 59  354 

0.  40  646 

9.  96  893 

26 

9.56  279 

9.59  391 

0. 40  009 

9.  90  888 

27 

9.56  311 

9.  59  429 

0.  40  571 

9.  96  883 

28 

9.60  343 

9.  59  466 

0.40  534 

9.  96  878 

29 

9.66  375 

33 

9.  59  503 

37 

0. 40  497 

9.96  873 

30 

9.  56  -408 

9.59  540 

0. 40  460 

9.  90  868 

31 

9.56  440 

9.  59  577 

0.40  423 

9.  96  863 

32 

9.  56  472 

9.  59  614 

0. 40  386 

9.  96  858 

33 

9.  56  504 

9.  59  651 

0.  40  349 

9.  90  853 

34 

9.  56  630 

32 

9.  59  688 

37 

0,40  312 

9.  96  848 

35 

9.56  568 

9.  59  725 

0.  40  275 

9.  96  843 

36 

9.56  599 

9.  59  762 

0.40  238 

9.96  838 

37 

9.56  631 

9.  69  799 

0.40  201 

9.  90  833 

38 

9.56  663 

9.  59  835 

0.  40  165 

9. 96  828 

39 

9.56  695 

32 

9.  69  872 

:i7 

0.  40  128 

9.  96  823 

10 

9.  56  727 

9.59  909 

0.  40  091 

9.  96  818 

41 

9.  56  759 

9. 59  946 

0.40  054 

9.  96  813 

42 

9.56  790 

9.  59  983 

0. 40  017 

9.96  808 

43 

9.56  822 

9.60  019 

0.39  981 

9.  90  803 

44 

9. 56  854 

32 

9.  60  056 

37 

0.  39  944 

9. 96  798 

45 

9. 56  886 

9.  60  093 

0.  39  907 

9.  96  793 

46 

9.56  917 

9.  60  130 

0. 39  870 

9.  96  788 

47 

9.56  949 

9.60  166 

0.  39  834 

9.  90  783 

48 

9.  56  980 

9.60  203 

0.  39  797 

9.  90  778 

49 

9.  57  012 

32 

9.  60  24U 

36 

0.  39  760 

9.  90  772 

50 

9.  57  044 

9.  60  276 

0.  39  724 

9.96  767 

51 

9.  57  075 

9.00  313 

0.  39  687 

9.  96  762 

52 

"  9.  57  107 

32 

9.  60  349 

0.  39  651 

9.96  757 

53 

9.  57  138 

9. 60  386 

0.  39  614 

9.  96  762 

54 

9.57  169 

32 

9.  60  422 

37 

0.  39  578 

9,  96  747 

55 

9.57  201 

9. 60  459 

0.39  541 

9. 96  742 

56 

9.57  232 

31 

9. 00  495 

0.  39  505 

9.  96  737 

57 

9.  57  264 

9.  60  532 

0.  39  468 

9.  96  732 

58 

9.  57  295 

9.60  508 

0.  39  432 

9. 96  727 

59 

9.  57  326 

31 

9.  60  005 

30 

0.  39  395 

9,90  722 
9,  90  717 

60 

9.  57  358 

9. 00  041 

0. 39  359 

L.  Cos. 

d. 

L.  Cotg. 

d.  c. 

L.  Tang. 

L.  Sin. 

38 
0,0 

37 

0,0 

1,3 

1,2 

1,9 

1,8 

2,5 

2,5 

3,2 

3,1 

3,8 

3,7 

4,4 

4,3 

5,1 

4,9 

6,7 

5,0 

6,3 

0,2 

12,7 

12,3 

19,0 

18,5 

25,3 

24,7 

31,7 

30,8 

33 

33 

0,6 

0,5 

1,1 

1,1 

1,0 

1,0 

2,8 

2,7 

3,3 

3,2 

3,8 

3,7 

4,4 

4,3 

5,0 

4,8 

5,5 

5,3 

11,0 

10,7 

10,5 

10,0 

22,0 

21,3 

2- ,5 

20,7 

0 

0,1 

0,1 

0,2 

0,2 

0,3 

0,2 

0,4 

0,3 

0,5 

0,4 

0,6 

0,5 

0,7 

0,6 

0,8 

0,7 

0,9 

0,8 

1,0 

0,8 

2,0 

1,7 

3,0 

2,5 

4,0 

3,3 

5,0 

6 

» 

37 

38 

3,1 

3,8 

9,2 

11,4 

15,4 

19,0 

21,6 

26,6 

27,8 

34,2 

33,9 

5 

4 

36 

88 

3,6 

4,8 

10,8 

14,2 

18,0 

23,8 

25,2 

33,2 

32,4 

3,0 
3,6 
4,2 
4,8 
5,4 
6,0 
12,0 
18,0 
24,0 
30,0 


2,1 
2,6 
3,1 
3,6 
4,1 
4,6 
5,2 
10,3 
15,6 
20,7 
25,8 


11,1 

18,5 
26,9 


13,9 
23,1 
32,4 


68<: 


276 


A  MANUAL  OF  TOrOGEAPHIC  METHODS. 


'1\UJLE  XXXVI.— io.'/"»'i'/""'(-  xine.f!,  cosines,  langciits,  ami  cu'tiiiflCH/.s— Coutiuiicd. 
[Extractwl  tVmii  Gauss'  Ldsaritliniii'  and  Triaiiuomotric  Tables.] 

22° 


LOGAEITHMS  OF  CIRCULAE  FUNCTIONS. 


277 


Table  XXXVI. — Logarithmic  sinesj  cosines^  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  liOgarithmic  and  Trigonometric  Tables.] 

23° 


L.Sm. 


9.59  188 
9.  59  218 
9.  59  247 
9.  59  277 
9.  59  307 


9.  59  330 
9.  59  366 
9. 59  396 
9.  59  425 
9. 59  455 


9.  59  484 
9.  59  614 
9.  59  543 
9.  59  573 
9.  59  602 


9.  59  632 
9.  59  661 
9.59  690 
9. 59  720 
9.  59  749 


9.  59  778 
9. 59  808 
9.  59  837 
9.  59  866 
9.  59  895 


9.  59  924 
9.  59  954 
9.59  983 
9.  60  012 
9.  60  041 


9. 60  215 
9.  60  244 
9. 60  273 
9. 60  302 
9.60  331 


9.60  359 
9.60  388 
9.  60  417 
9. 60  446 
9.60  474 


9.  60  503 
9. 60  532 
9. CO  561 
9. 60  589 
9.  60  618 


9. 60  646 
9. 60  675 
9. 60  704 
9.  60  732 
9.  60  761 


9.  60  789 
9.  60  818 
9.  60  846 
9.60  875 
9.  60  903 
9.60  931 


9.  62  785 
9.  62  820 
9.  62  855 
9. 62  890 
9.  62  926 


9.  62  961 
9.  62  996 
9.63  031 
9.  63  066 
9.  63  101 


9.  63  135 
9.  63  170 
9.63  205 
9.  63  240 
9.  63  275 


9.  63  310 
9.  63  .345 
9.63  379 
9. 63  414 
9.  63  449 


9.63  484 
9.  63  519 
9.  63  553 


9.  63  726 
9.63  761 
9.  63  796 


9.  64  003 
9.  64  037 
9.  64  072 
9.  64  106 
9.  64  140 


9.  64  175 
9. 64  209 
9.  64  243 
9.64  278 
9.64  312 


9.  64  346 
9.  64  381 
9.  64  415 
9.  64  449 
9. 64  483 


9.  64  517 
9.  64  552 
9. 64  586 
9.  64  620 
9.64  654 


9.  64  688 
9.  64  722 
9. 6i  756 
9. 64  790 
9.  64  824 
9.  64  858 


L.  Cotg.   a.  c. 


L.  Cotg. 


0.  37  215 
0.37  180 
0.  37  145 
0.37  110 
0.  37  074 


0.  37  039 
0.  37  004 
0.  36  969 
0.  36  934 
0.  36  899 


0.36  865 
0.30  8.30 
0.36  795 
0.  36  760 
0. 36  725 


0.36  690 
0,  ,36  655 
0.  36  621 
0.  36  586 
0.30  551 


0.  36  516 
0.  36  481 
0.  36  447 
0.36  412 
0.  36  377 


0. 36  343 
0.  36  308 
0.  36  274 
0.  36  239 
0.  36  204 


0. 36  170 
0.  36  135 
0.  36  101 
0.  36  066 
0.  36  032 


0. 35  997 
0.35  963 
0. 35  928 
0. 35  894 
0.  35  860 


0.  35  825 
0.35  791 
0.  35  757 
0.  35  722 
0.35  688 


0.35  654 
0.35  619 
0. 35  585 
0.35  551 
0.35  517 


0.  35  483 
0.  35  448 
0.  35  414 
0.  35  380 
0.  35  346 


0.  35  312 
0.  35  278 
0.  35  244 
0.  35  210 
0. 35  176 
0.  35  142 


9.  96  403 
9.96  397 
9.  96  392 
9.  96  387 
9.  96  381 


9.96  376 
9.  96  370 
9.96  365 
9.  96  360 
9.  96  354 


9. 96  349 
9.96  343 
9.90  338 
9.96  333 
9.96  327^ 
9.96  322 
9. 96  316 
9.96  311 
9.96  305 


9. 96  284 
9. 96  278 
9.96  273 


9.90  207 
9.  96  262 
9.  96  256 
9.  96  251 
9.  96  245 


9.96  207 
9.  96  201 
9.96  196 
9.96  190 


9.96  185 
9.96  179 
9.  96  174 
9.  96  168 
9.96  162 


9.96  157 
9.  96  151 
9  96  146 
9.96  140 
9.96  135 


9.  90  129 
9.96  123 
9.96  lis 
9.96  112 
9.96  107 


9.96  101 
9. 96  095 
9.  96  090 
9.  96  084 
9.96  079 
9.  96  073 


36 

0,6 

35 

0,6 

1/2 

1,2 

1,8 

1,8 

2,4 

2,3 

3,0 

2,9 

3,6 

3,5 

4,2 

4,1 

4,8 

4,7 

5,4 

5,2 

6,0 

5,8 

12,0 

11,7 

18,0 

17,5 

24,0 

23,3 

30,0 

29,2 

30 

29 

0,5 

0,5 

1,0 

1,0 

1,5 

1,4 

2,0 

1,9 

2,5 

2,4 

3,0 

2,9 

3,5 

3,4 

4,0 

3,9 

4,5 

4,4 

5,0 

4,8 

10,0 

9,7 

15,0 

14,5 

20,0 

19,3 

25,0 

24,2 

3,4 
4,0 
4,5 
5,1 
5,7 
11,3 
17,0 
22,7 
28,3 


1,9 
2,3 
2,8 
3,3 
3,7 
4,2 
4,7 
9,3 
14,0 
18,7 
23,3 


G 

6 

36 

35 

3,0 

2,9 

9,0 

8,8 

15,0 

14,6 

21,0 

20,4 

27,0 

26,2 

33,0 

32,1 

8,5 
14,2 
19,8 
25,5 
31,2 


3,5 

3,4 

10,b 

10,2 

IV, b 

17,0 

24,5 

23,8 

31,0 

30,6 

66= 


278 


A  MANUAL  OF  TOPOGKAPHIC  METHODS. 


Table  XXXVI. — Logarithmic  sines,  cosines,  tatigcnts,  and  cotanyenis — Contimied. 

[Exti-actecl  from  Gauss'  Logaritlimic  and  Trigonometric  Tables.) 

24° 


9. 60  960 
9.  60  988 
9.  61  QIC 

9.61  045 


9.  61  073 
9. 61  101 
9.61  129 
9.61  loS 
9.61  186 


9.61  214 
9.  01  242 
9.61  270 
9.61  298 
9.61  326 


9.61  354 
9.61  382 
9.61  411 
9.61  438 
9.  61  466 


9.  61  494 
9.  61  522 
9.61  550 
9.61  578 
9.61  606 


9.61  634 
9.61  662 
9.6]  689 
9.  61  717 
9.  61  745 


9.61  773 
9.61  800 
9. 61  828 
9.  61  856 
9.61  883 


9.  61  911 
9.61  939 
9.61  966 
9. 61  994 
9.  62  021 


9.  02  049 
9.  62  076 
9.  62  104 
9.  62  131 
9. 62  159 


9.62  186 
9. 62  214 
9.  62  241 
9.  62  268 
9.  62  296 


9.  62  323 
9. 62  350 
9.  62  377 
9.  62  405 
9.  62  432 


9.  62  459 
9.  62  486 
9.  62  513 
9. 62  541 
9. 62  568 
9.  62  595 


9.  64  858 
9.  04  892 
9.  64  026 
9.  64  960 
9.  64  994 


9.  65  028 
9.  65  062 


9.65  197 
9.  65  231 
9.  65  265 
9.  65  299 
9. 65  333 


9.  65  366 
9. 65  400 
9.  65  434 
9. 63  467 
9.  65  501 


9. 65  535 
9.  65  568 
9.  65  602 
9.  65  636 
9.  65  669 


9. 65  703 
9.  65  736 
9. 65  770 
9.  65  803 
9.  65  837 


9.  65  870 
9. 65  904 
9.  65  937 
9.  65  971 
9.  66  004 


9.  66  038 
9.  66  071 
9. 06  104 
9. 66  138 
9.  66  171 


9.  66  204 
9-.  66  238 
9.  66  271 
9.66  304 
9.  66  337 


9.  66  371 
9. 66  404 
9.  66  437 
9.  66  470 
9.  66  503 


9.  66  .537 
9.  66  570 
9.  66  603 
9.  66  636 
9.  66  669 


9.  66  702 
9.  66  735 
9.  66  768 
9. 66  801 
.  9.  66  834 
9.  66  867 


L.  Cotg. 


0.  35  142 
0.35  108 
0.  35  074 
0.  35  040 

0. 35  oon 


0.  34  972 
0.  34  938 
0. 34  904 
0.  34  870 
0. 34  836 


0.  34  803 
0.34  769 
0. 34  735 
0.  34  701 
0.34  667 


0.34  634 
0.34  600 
0.  34  566 
0. 34  533 
0.  34  499 


0.  34  465 
0. 34  432 
0.  34  398 
0.  34  364 
0.  34  331 


0.  34  297 
0. 34  264 
0.  34  230 
0. 34  197 
0.34  163 


9. 96  073 
9. 96  067 
9. 96  062 
n.  96  056 
9.  96  050 


9.  96  045 
9. 96  039 
9.  96  034 
9.  96  023 
9.96  022 


9.  96  017 
9.96  Oil 
9.  96  005 
9.  96  000 
9. 95  994 


9.  95  988 
9. 95  982 
9. 95  977 
9. 95  971 
9.  95  965 


9.  95  960  I 
9.  95  954 
9.  95  948 
9. 95  942 
9.  96  937 


0.  34  130 
0.  34  096 
0.  34  063 
0.  34  029 
0.33  996 


0.33  962 
0.33  929 
0.33  896 
0.33  802 
0.  33  829 


9.  95  931 
9.  95  925 
9.  95  920 
9.95  914 
9.95  908 


9.95  902 
9.95  897 
9. 95  891 
9.  95  885 
9.  95  879 


0.  33  796 
0. 33  762 
0.  33  729 
0. 33  696 
0.  33  663 


0.1 


629 


9. 95  873 
9.  95  868 
9.  95  862 
9.  95  856 
9.  95  850 


9.  95  844 
9.  95  839 
9.  95  833 


0.33  596 
0. 33  563  I 


0.  33  463 
0.  33  430 
0.  33  397 
0. 33  364 
0.33  331 


0. 33  298 
0. 33  265 
0. 33  232 
0. 33  199 
0.  33  166 
0. 33  133 


9.  95  810 
9.95  804 
9. 95  798 
9. 95  792 


9.  95  757 
9.-95  751 
9.  95  745 
9,  95  739 
9.95  733 
9.  95  728 


0,6 

0,6 

1,1 

1,1 

i,v 

1,6 

2,3 

2,2 

2,8 

2,8 

3,4 

3,3 

4,0 

3,8 

4,b 

4,4 

5,1 

0,0 

5,V 

b,6 

11,3 

11,0 

17,0 

16,b 

22,7 

22,0 

28,3 

2V,b 

29 

28 

0,5 

0,5 

1,0 

0,9 

1,4 

1,4 

1,9 

1,9 

2'4 

2,3 

2,9 

2,8 

3,4 

3,3 

3,9- 

3,7 

4,4 

4,2 

4,8 

4,7 

9,7 

9,3 

14,5 

14,0 

19,3 

18,7 

24,2 

23,3 

G 

6 

34 

S3 

2,8 

2,8 

8,5 

8,2 

14,2 

13,8 

19,8 

19,2 

25,5 

24,8 

31,2 

30,2 

0,4 
0,9 

1;4: 

1,8 
2,2 
2,7 
3,2 
3,6 
4,0 
4,5 
9,0. 
13,5 
18,0 
22,5 


10,2 
17,0 
23,8 
30,6 


63"^ 


LOGARITHMS  OF  CIRCULAR  FUNCTIONS. 


279 


Tablk  XXXVI. — Logarithmic  Hnes,  cosi^ies,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometrii;  Tables.] 

950 


9.  62  595 
9.  62  622 
9.  62  649 
9.  62  676 
9.  62  703 


9.62  7TO 
9.  62  757 
9.  62  784 
9.62  811 
9.  62  838 


9.62  865 
9.  62  892 
9.62  918 
9.62  945 
9.62  072 


9. 62  999 
9.  63  026 
9.  63  052 
9.  03  079 
9.  63  106 


9.03  133 
9. 63  159 
9.63  180 
9.  63  213 
9.  63  239 


9. 63  266 
9.63  292 
9.  63  319 
9.63  345 
9. 63  372 


9.  63  398 
9. 63  425 
9.  63  451 
9.  63  478 
9.  63  304 


9.  63 .531 
9.  63  557 
9.63  583 
9.  63  610 
9.  63  636 


9.  63  662 
9.  (i3  689 
9.63  715 
9.  63  741 
9.63  767 


9.  63  794 
9.63  820 
9.  63  846 
9.  63  872 
9.  63  898 


9.63  924 
9. 63  950 

9. 63  976 
9.  64  002 

9. 64  028 
9.  64  054 
9.  64  080 
9.  64  106 
9.  64  132 
9.  §4  158 
9.  64  184 


9.  66  867 
9. 66  900 
9.  66  933 
9. 66  966 
9. 66  999 


9.07  032 
9.67  065 
9. 67  098 
9.  67  131 
9.  67  163 


9.67  196 
9.  67  229 
9.  67  262 
9.  67  293 
9.  67  327 


9.  67  360 
9.  67  393 
9.  67  426 
9.  67  458 
9.  67  491 


9.  67  524 
9.  67  556 
9.67  589 
9.67  622 
9.  67  654 


9.67  687 
9.  67  719 
9.  67  752 
9.  67  785 
9.67  817 


9.  67  850 
9.  67  882 
9.  67  915 
9.  67  947 
9.  67  980 


9.  68  012 
9.  68  044 
9.  68  077 
9.68  109 
9.  68  142 


9.  68  497 
9.  68  529 
9.  68  561 
9.  68  593 
9. 68  626 


L.  Cotg. 


L.  Cotg. 


0.33  133 
0.33  100 
0.33  067 
0.33  014 
0.33  001 


0.  32  968 
0.  32  935 
0.  32  902 
0.  32  869 
0. 32  837 


0.  32  804 
0. 32  771 
0.  32  738 
0. 32  705 
0. 32  673 


0. 32  640 
0. 32  607 
0.  32  574 
0.  32  542 
0. 32  509 


0.  32  476  f 
0. 32  444 
0.32  411 
0. 32  378 
0.  32  346 


0.32  313 
0. 32  281 
0.  32  248 
0.  32  215 
0.32  183 


0.  32  150 
0.  32  118 
0. 32  085 
0. 32  053 
0. 32  020 


0.  31  988 
0.31  9.56 
0.31  923 
0.31  891 
0.31  858 


0.31  826 
0.  31  794 
0.31  761 
0.  31  729 
0.31  697 


0.31  664 
0.31  632 
0.31  600 
0.31  568 
0.31  535 


0.  31  503 
0.  31  471 
0.31  439 
0.31  407 
0.31  374 


0.31  342 
0.31  310 
0.  31  278 
0. 31  246 
0.31  214 
0. 31  182 


9.95  728 
9.  95  722 
9.95  716 
9. 95  710 
9.  95  704 
9795^8" 
9.95  692 
9.  95  686 
9.  95  680 
9.  95  674 


9.  95  668 
9. 95  663 
9.95  657 
9.95  65] 
9.  95  645 


9.  95  639 
9.95  633 
9.95  627' 
9. 95  621 
9.95  615 


9. 95  603 
9. 95  597 
9. 95  591 
9.95  585 


9.  95  679 
9.  95  573 
9.95  567 
9.  95  661 
9.  95  555 


9.  95  549 
9.  95  543 
9.  95  537 
9.  95  531 
9.  95  525 


9. 95  519 
9. 95  513 
9.  95  507 
9.  95  500 
9.  95  494 


9.  95  488 
9. 95  482 
9.  95  476 
9. 95  470 
9. 95  464 


9.95  4.58 
9.  95  453 
9.  95  446 
9.  95  440 
9.  95  434 


9.  95  427 
9.  95  421 
9.95  415 
9.  95  409 
9.95  403 


9.  95  397 
9.  95  391 
9.  95  384 
9.  95  378  I 
9. 95  372 
9. 95  366 


5,0 

4,8 

5,0 

5,3 

11,0 

10,7 

16,b 

16,0 

22,0 

21,3 

2V,b 

26,7 

27 

26 

0,4 

0,4 

0,9 

0,9 

1,4 

1,3 

1,8 

1,7 

2,2 

2,2 

2,7 

2,0 

3,2 

3,0 

3,6 

3,5 

4-0 

3,9 

4,b 

4'3 

9,0 

8,7 

13,5 

13,0 

18,0 

17,3 

22,5 

21,7 

0,1 

6 

0,1 

0,2 

0,2 

0,4 

0,3 

0,5 

0,4 

0,6 

0,5 

0,7 

0,6 

0,8 

0,7 

0,9 

0,8 

1,0 

0,9 

1,2 

1,0 

2,3 

2,0 

3,5 

3,0 

^,^ 

4,0 

5,8 

5,0 

' 

G 

32 

32 

2,3 

2,7 

6,9 

8,0 

11,4 

13,3 

16,0 

18,7 

20,6 

24,0 

25,1 

29,3 

29,7 

9,9 
16,5 
23,1 
29,7 


64^^ 


280 


A  MANUAL  OF  TOrOGEAPHIC  METHODS. 


Table  XXXVI. — Loijariihmic  shies,  cosi-nes,  tmufenis,  and  cotangents — Contin;ied. 
[Extracted  I'rom  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

26° 


9.64  184 
9.  64  210 
9.  64  236 
9. 64  262 
9.  64  288 


9.  64  313 
9.64  339 
9.  64  365 
9.C4  391 
9.  64  417 


9.  64  442  1 
9.  64  468 
9.  64  494  1 
9.  64  519 
9.  64  545 


9.  64  571 
9.64  596 
9. 64  622 
9.  64  647 
9.  64  673 


9.  64  698 
9.  61  724 
9. 64  749 
9.  64  775 
9. 64  800 


9.  64  826 
9.  64  851 
9.  64  877 
9.  64  902 
9.  64  927 
9.  64  953 

9.64  978 
9.  65  003 

9. 65  029 
9.  65  054 


9.65  331 
9.65  356 
9.  65  381 
9.65  406 
9.  65  431 


9.65  456 
9. 65  481 
9.65  506 
9.  65  531 
9.65  556 


9. 65  580 
9.  65  605 
9.  65  630 
9.  65  655 
9.65  680 
9.  65  705 


9.  68  850 
9.68  882 
9.  68  914 
9.68  946 


9.  68  '978 
9.69  010 
9.09  042 
9.69  074 
9.69  106 


9.  69  138 
9.69  170 
9.69  202 
9.69  234 
9.69  266 


9.  69  298 
9.  09  329 
9.  69  361 
9.  69  393 
9.  69  425 


9.70  089 
9.70  121 
9. 70  152 
9.70  184 
9.70  215 


•  9. 70  247 
9.70  278 
9.  70  309 
9. 70  341 
9.70  372 


9.  70  404 
9. 70  435 
9. 70  466 
9.  70  498 
9. 70  529 


L.  Cotg. 


0.  31  182 
0.31  150 
0.31  118 
0.31  086 
0.31  054 


0.31  022 
0.30  990 
0.30  958 
0.30  926 
0.  30  894 


0.30  S62 
0.30  830 
0.  30  798 
0.30  766 
0.30  7.34 


0.30  702 
0.30  671 
0. 30  639 
0. 30  607 
0. 30  575 


0. 30  385 
0.  30  353 
0.  30  321 
0. 30  290 
0. 30  258 


0.  30  226 
0.  30  195 
0.30  163 
0.  30  132 
0.  30  100 


0.  30  (168 
0. 30  037 
0.  30  005 
0.  29  974 
0,  29  942 


0.29  911 
0. 29  879 
0.  29  848 
0.  29  816 
0.  29  785 


0.29  753 
0.  29  722 
0.  29  691 
0.  29  659 
0.  29  628 


0.  29  596 
0.29  565 
0.  29  534 
0.  29  502 
0. 29  471 


0. 29  440 
0.29  408 
0. 29  377 
0.  29  346 
0.  29  315 
0. 29  283 


9.95  360 
9.  95  360 
9.95  354 
9.95  348 
9.95  341 
9.95  335 
9.95  329 
9.95  323 
9.95  317 
9.95  310 


9.05  304 
9.95  298 
9.  95  292 
9.95  286 
9.95  279 


9.95  273 
9.  95  267 
9.  95  261 
9.95  254 
9.  95  248 


9.95  242 
9.95  236 
9. 95  229 
9.  95  223 
9. 95  217 


9. 95  211 
9. 95  204 
9. 95  198 
9.  95  192 
9.95  185 


9.95  179 
9.95  173 
9.  95  167 
9.  95  160 
9.95  154 


9. 95  148 
9.95  141 
9.95  135 
9.95  129 
9.95  122 
9.  95  lie 
9.95  110 
9.  95  103 
9.  95  097 
9.95  090 


9.  95  084 
9.  95  078 
9.  95  071 
9.95  065 
9.95  059 


9.  95  052 
9.95  046 
9. 95  039 
9.  95  033 
9.95  027 


9.  95  020 
9.  95  014 
9.  95  007 
9.05  001 
9  94  995 
9.  94  988 


d. 

«0 

6 

59 

58 

57 

V 

56 

6 

55 

54 

53 

52 

7 

51 

6 

50 

6 

49 

48 

6 

47 

6 

46 

45 

44 

43 

42 

41 

6 

40 

39 

V 

38 

37 

36 

6 

"35 

34 

6 

33 

32 

31 

6 

SO 

29 

28 

6 

26 

6 

25 

24 

6 

23 

B 

22 

21 

6 

2(1 

19 

V 

18 

6 

17 

V 
6 

16 

15 

14 

13 

b 

12 

7 
6 

11 

10 

9 

8 

7 

6 

5 

6 

4 

3 

2 

6 

1 

V 

0 

(1. 

' 

32 

1 

0,5 

2 

1,1 

3 

1,6 

4 

2,1 

5 

2,7 

6 

3,2 

7 

3,7 

8 

4,3 

9 

4,8 

1» 

5,3 

^(1 

10,7 

30 

16,0 

40 

21,3 

i>0 

26,7 

10,3 
15,5 
20,7 
25,8 


20 

25 

0,4 

0,4 

0,9 

0,8 

1,3 

1,2 

1,7 

1,7 

2,2 

2,1 

2,6 

2,5 

3,0 

2,9 

8,5 

3,3 

3,9 

3,8 

4,3 

4,2 

8,7 

,      8,3 

13,0 

12,5 

17,3 

16,7 

21,7 

20,8 

0,8 
1,2 
1,6 
2,0 
2,4 
2,8 
3,2 
3,0 
4,0 
8,0 
12,0 
16,0 
20,0 


_ 

33 

31 

2,3 

2,2 

6,9 

6,0 

11,4 

11,1 

16,0 

15,5 

-20,6 

19'9 

25,1 

24,4 

29,7 

.28,8 

2,7 
8,0 
13,3 
18,7 
24,0 


63° 


LOGAEITHMS  OF  CIECULAE  FUNCTIONS. 


281 


Table  XXXVI. — Lofiarithmic  sines,  cosines,  tangents,  and  cotangents — Continued. 

fExtracted  from  Gauss'  Logaritlimic  anil  Trifcononietric  Tables.] 

27° 


9.  65  705 
9.  65  729 
9. 65  754 
9.  65  779 
9. 65  804 


9.  65  828 
9.  65  853 
9. 65  878 
9. 65  902 
9.  65  927 
9.65  952 
9.  65  976 
9.  66  001 
9.  66  025 
9.  66  050 


■  9.  66  U75 
9.  66  099 
9. 66  124 
9.66  148 
9.66  173 


9.66  197 
9.  66  221 
9.66  246 
9. 66  270 
9.  66  295 


9.66  319 
9. 66  343 
9.  66  368 
9.  66  392 
9,  66  416 


9.  66  441 
9.  66  465 
9.  66  489 
9.66  513 
9.66  537 


9.66  562 
9. 66  586 
9.  66  610 
9.66  634 
9. 66  658 


9.66  I 


1  706 


9.  66  S03 
9.  66  827 
9. 66  851 
9.  66  875 
9. 66  899 


9.  66  922 
9.  66  946 
9.66  970 
9.  66  994 
9.  67  018 


9.70  779 
9.70  810 
9.70  841 
9.  70  873 
9. 70  904 
9.70  935 
9. 70  966 
9.  70  997 


9.71  028 
9.71  059 
9.71  090 
9.71  121 
9.71  153 


9.71  493 
9.  71  524 
9.71  555 
9. 71  586 
9.71  617 


9.71  648 
9.71  679 
9. 71  709 


9. 72  262 
9.  72  293 
9.72  323 
9.72  354 
9. 72  384 
9.  72  415 
9.72  445 
9. 72  476 
9.  72  506 
9. 72  537 
9. 72  567 


L.  Cotg. 


L.  Cotg. 


0. 29  283 
0.  29  252 
0.29  221 
0.29  190 
0.29  159 


0.29  127 
0.29  096 
0.  29  065 
0.  29  034 
0.  29  003 


0.28  972 
0.  28  941 
0.28  910 
0.28  879 
0.28  847 


0.28  816 
0. 28  785 
0.  28  754 
0.28  723 
J)^28  692 
0.  28  601 
0.28  630 
0.  28  599 
0.28  569 
0. 28  538 


0.28  507 
0. 28  476 
0.  28  445 
0.  28  414 
0. 28  383 


0. 28  352 
0. 28  321 
0.  28  291 
0.  28  260 
0.  28  229 


0.28  198 
0.28  167 
0.  28  137 
0.  28  106 
0.28  075 


0.  28  045 
0.  28  014 
0. 27  983 
0.27  952 
0.27  922 


0.  27  891 
0.  27  860 
0.  27  830 


0.  27  707 
0.27  677 
0. 27  646 
0. 27  616 
0. 27  585 
0. 27  555 
0.  27  524 
0.  27  494 


9.  94  988 
9. 94  982 
9.94  975 
9.94  969 
9.94  962 


9.  94  956 
9.94  949 
9.94  943 
9.  94  936 
9.  94  930 


9. 94  923 
9.  94  917 
9.  94  911 
9.  94  904 
9. 94  898 


9.94  891 
9.94  885 
9.  94  878 
9.  94  871 
9.94  865 


9.94  858 
9.  94  852 
9.  94  845 
9.94  839 
9.  94  832 


9.94  826 
9. 94  819 
9.94  813 
9.94  806 
9.  94  799 


9. 94  793 
9.  94  786 
9. 94  780 
9.  94  773 
9.  94  767 


9.94  760 
9.  94  763 
9. 94  747 
9.  94  740 
9.94  734 


9.  94  727 
9. 94  720 
9.  94  714 
9.  94  707 
9.  94  700 


9.  94  660 
9.  94  654 
9.  94  647 
9. 94  640 
9.  94  634 
9.  94  627 
9.  91  620 
9.  94  614 
9.  94  607 
9.  94  600 
9.  94  593 


32 

31 

0,5 

0,5 

1,1 

1,0 

1,6 

1,6 

2,1 

2,1 

2,7 

2,6 

3,2 

3,1 

3,7 

3,6 

4,3 

4,1 

4,8 

4,6 

5,3 

5,2 

ig,7 

10,3 

16,0 

15,5 

21,3 

20,7 

26,7 

25,8 

25 

24 

0,4 

0,4 

0,8 

0,8 

1,2 

1,2 

1,7 

1,6 

2,1 

2,0 

2,5 

2,4 

2,9 

2,8 

3,3 

3,2 

3,8 

3,6 

4,2 

4,0 

8,3 

8,0 

12,5 

12,0 

16,7 

16,0 

20,8 

20,0 

10,0 
15,0 
20,0 
25,0 


2,3 
2,7 
3,1 
3,4 
3,8 
7,7 
11,5 
15,3 
19,2 


' 

G 

30 

31 

2,1 

2,6 

6,4 

7,8 

10,7 

12,9 

15,0 

18,1 

19,3 

23,2 

23,6 

28,4 

27,9 

2,5 
7,5 
12,5 
17,5 
22,5 
27,5 


62= 


282 


A  MANUAL  OF  TOPOGKAPHIC  METHODS. 


Table  XXKri.— Logarithmic  sines,  cosines,  tangents,  and  coinngents—ContiaueA. 

[Extracted  from  Gauss'  Logai-ithmic  and  Trigonometric  Tables.] 

S8° 


SI 

•0,5 

30 

0,5 

1,0 

1,0 

1,0 

1,5 

2,1 

2,0 

2,6 

2,5 

3,1 

3,0 

3,6 

3,5 

4,1 

4,0 

4,6 

4,5 

5,2 

5,0 

10,3 

10,0 

15,5 

15,0 

20,7 

20,0 

25,8 

25,0 

24 

2S 

0,4 

0,4 

0,8 

0,8 

1,2 

1,2 

■  1,6 

2,0 

1,9 

2,4 

2,3 

2,8 

2,7 

3,2 

3,1 

3,6 

3,4 

4,0 

3,8 

8,0 

7,7- 

12,0 

11,5 

10,0 

15,3 

20,0 

19,2 

1,4 
1,9 
2,4 
2,9 
3,4 
3;9 
4,4 
4,8 
9,7 
14,5 
19,3 
24,2 


11,0 
14,7 
18,3 


6 

SI 

31 

2,2 

2,6 

6,6 

7,8 

11,1 

12,9 

15,5 

18,1 

19,9 

23,2 

24,4 

28,4 

28,8 

2,5 
7,5 
12,5 
17,5 
23,5 
27,5 


61° 


LOGARITHMS  OF  CIRCULAE  FUNCTIONS. 


283 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logaritlimic  and  Trigonometric  Tables.] 

39° 


L.  Tang. 


9. 68  G71 
9. 68  694 
9.68  716 
9.  68  739 
9^68  762 
9.  68  784 
9.  68  807 
9.  68  829 
9.68  852 
9.68  875 
'9.  68  897" 
9.  68  920 
9.  68  942 
9.  08  965 
9.68  987 

9.  egliiir 

9.  69  032 
9.  69  055 
9.  69  077 
9.  69  100 
9.  69  122 


9.  69  345 
9.  69  368 
9  69  390 
9.  69  412 


9.69  611 
9.  69  633 
9.  69  655 


9. 74  375 
9. 74  405 
9.  74  435 
9.  74  465 
9. 74  494 


9.  74  524 
9. 74  554 
9. 74  583 
9.  74  613 
9.  74  643 


9. 74  969 
9.  74  998 

9.75  028 
9.  75  058 
9.75  087 


9.75  205 
9. 75  235 


9.  75  264 
9.  75  294 
9.  75  323 
9.  75  353 
9.75  382 


9.75  411 
9.  75  441 
9. 75  470 
9.75  500 
9.  75  529 


9.75  558 
-9.  75  588 
9.75  617 
9.  75  647 
9.  75  676 


9.75  764 
9.  75  793 
9. 75  822 


9.  75  852 
9.75  881 
9.  75  910 
9.  75  939 
9.  75  969 


9.75  998 
9.  76  027 
9.  76  056 


L.  Cotg. 


0.24  736 
0. 24  706 
0.  24  677 
0. 24  647 
0. 24  618 


0.  24  589 
0.  24  559 
0.  24  530 
0. 24  500 
0.  24  471 


9.93  934 
9.93  927 
9. 93  920 
9.93  912 
9.  93  905 


30 

0,5 

29 

0,5 

1,0 

1,0 

1,5 

lA 

2,0 

1,9 

2,5 

2,4 

3,0 

2,9 

3,5 

3,4 

4,0 

3,9  1 

4,5 

4,4 

5,0 

4,8 

10,0 

9,7 

15,0 

14,5 

20,0 

19,3 

25,0 

24,2 

2*2 

8 

0,4 

0,1  ' 

0,7 

0,3 

1,1 

0,4 

1,5 

n,5 

1,8 

0,7 

2,2 

0,8 

2,6 

0,9 

2,9 

1,1 

3,3 

1,2 

3,7 

1,3 

7,3 

2,7 

11,0 

4,0 

14,7 

5,3 

18,3 

6,7 

8 

S 

30 

29 

1,9 

1,8 

5,6 

5,4 

9,4 

9,1 

13,1 

12,7 

16,9 

16,3 

20,6 

19,9 

24,4 

23,6 

28,1 

27,2 

2,1 

2,1 

6,4 

0,2 

10// 

10,4 

15,0 

14,5 

19,3 

18j6 

23.6 

22,8 

27,'J 

26,9 

11,5 
15,3 
19,2 


60= 


284 


A  MANUAL  OF  TOrOGKAPHlO  METHODS. 


Table  XXXVI. — Lognrifhmic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

30° 


9. 09  919 
9.  69  941 
9. 69  963 
9.  69  984 


9.70  OUO 
9.  70  02S 
9.  70  050 
9.  70  072 
9.70  093 


9.70  115 
9. 70  137 
9.70  159 
9.70  189 
9. 70  202 
9.  70  22T 
9.  70  245 
9. 70  267 
9.  70  288 
9. 70  310 


9.  70  332 
9. 70  353 
9.  70  375 
9. 70  396 
9. 70  418 


9.70  654 
9.  70  675 
9. 70  697 
9.  70  718 


9.  70  909 
9.  70  931 
9.70  952 


9. 76  144 
9.  76  173 
9.  76  202 
9. 76  231 
9.76  261 


9. 76  290 
9.76  319 
9.76  348 
9. 76  377 
9.76  406 


9.76  435 
9. 76  464 
9.  76  493 
9. 76  522 
9. 76  551 


9.77  159 
9.77  188 
9.77  217 
9.77  240 
9.  77  274 


9.77  303 
9. 77  332 
9.77  361 
9. 77  390 
•9.77  418 


9. 77  447 
9.  77  476 
9.  77  505 
9. 77  533 
9.  77  562 


9.  77  591 
9. 77  619 
9.77  648 
9. 77  677 
9.77  706 


9.77  734 
9.  77  763 


L.  Cotg. 


0.  23  856 
0.  23  827 
0.  23  798 
0.23  769 
0.23  739 


9.  93  753 
9.  93  746 
9.  93  738 
9.93  731 
9.93  724 


0.  23  710 
0.23  681 
0.23  652 
0.23  623 
0.23  594 


0, 23  565 
0.23  536 
0.23  .507 
0.23  478 
0.  23  449 


9.93  717 
9.  93  709 
9.93  702 
9.93  695 
J)^93  687 
9.93  680 
9.93  673 
9.93  065 
9.93  058 
9.93  050 


0. 23  420 
0.23  391 
0. 23  361 
0.23  332 
0.23  303 


9.93  643 
9.93  636 
9.93  628 
9.93  621 
9.  93  614 


0.  23  275 
0.23  246 
0.  23  217 
0.23  188 
0.23  159 


9.93  606 
9.93  599 
9.93  591 
9.93  584 
9.93  577 


0.23  130 
0.23  101 
0.23  072 
0.23  043 
0.23  014 


9.93  569 
9.  93  562 
9.93  554 
9.93  547 
9.93  539 


9.93  532 
9.93  525 
9.  93  517 
9.  93  510 
9.93  50£ 
9. 93  495 
9.  93  487 
9.  93  480 
9.93  472 
9.93  465 


0.  22  097 
0.  22  668 
0.22  639 
0.22  610 
0.22  582 


9.93  457 
9.93  450 
9.  93  442 
9.93  435 
9.93  427 


0.22  653 
0.  22  524 
0.  22  495 
0.  22  467 
0.  22  438 


9.  93  420 
9.93  412 
9.93  405 
9.  93  397 
9.93  390 


0. 22  409 
0.22  381 
0.22  352 
0. 22  323 
e.  22  294 


9. 93  382 
9.93  375 
9.93  367 
9. 93  360 
9.  93  352 


0. 22  266 
0. 22  237 
0.22  209 
0.  22  180 


L.  Tang. 


9.93  344 
9. 93  337 
9. 93  329 
9.  93  322 
9. 93  314 
9.  93  li07" 


80 

29 

0,5 

0,5 

1,0 

1,0 

1,5 

14 

2,0 

1,9 

2,5 

2,4 

3,0 

2,9 

3,5 

3,4 

4,0 

3,9 

4,5 

4,4 

5,0 

4,8 

io;o 

9,7 

15,0 

14,5 

20,0 

19,3 

25,0 

24,2 

2,8 
3,3 
3,7 
4,2' 
*,7 
9,3 
14,0 
18,7 
23,3 


1,0 
1,4 
1,8 
2,1 
2,4 
2,8 
3,2 
3,5 
7,0 
10,5 
14,0 
17,5 


; 

J 

30 

29 

2,1 

2,1 

6,4 

6,2 

10,7 

10,4 

15,0 

14,5 

19,3 

18,6 

23,6 

22,8 

27,9 

26,9 

10,0 
14,0 
18,0 
22,0 
26,0 


59° 


LOGAEITHMS  OF  GIRCULAE  FUNCTIONS. 


285 


Taule  XXXVI. — Logarithmic  sineSj  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logaritlimic  and  Trigonometric  Tables.] 

31° 


9.71  184 
9.71  205 
0.  Tl  2a6 
9. 71  247 
9.71  268 


9.71  289 
9.  71  310 
9.71  331 
9.  71  352 
9.  71  373 


9.71  393 
9.  71  414 
9.  71  435 
9.  71  456 
9.  71  477 


9. 71  498 
9.71  519 
9.  71  539 
9.  71  560 
9.  71  581 


9. 71  602 
9. 71  622 
fl.71  643 
9.  71  664 
9.71  685 


9.  71  705 
9.71  726 
9.71  747 
9.71  767 
9.  71  788 


9.  71  809 
9.  71  829 
9.71  850 
9.71  870 


9.71  911 
9.71  932 
9.71  952 
9,71  973 


9.  72  014 
9.  72  034 
9.72  055 
9.  72  075 
9.  72  096 


9. 72  238 
9. 72  259 
9.  72  279 
9.  72  299 


•8  249 

'&  277 


9.  78  306 
9.:  8  334 
9.  78  363 
9. 78  391 
9.  78  419 


8  448 
8  476 
9.  78  505 
9.78  533 
9.  78  562 


9.78  590 
9.78  018 
9.  78  647 
9.78  675 
9.  78  704 


9.  78  732 


9.7 


760 


9.  78  817 
9.  78  845 


9.  78  874 
9.  78  902 
9. 78  930 
9.78  959 
9.  78  987 


9.79  015 
9.79  043 
9.  79  072 
9.  79  100 
9.79  128 


9.  79  156 
9.  79  185 
9.  79  213 
9.  79  241 
9.  79  269 


9.  79  297 
9.  79  326 
9. 79  354 
9.79  382 
9.  79  410 


L.  Cotg. 


0.22  123 
0.  22  094 
0.  22  065 
0, 22  037 
0.  22  008 


0.21  980 

0.  21  951 

0.21  923 

0.21  894 

0.21  865 


0.  21  837 
0,21  808 
0. 21  780 
0.21  751 
0.21  723 


U. 21  694 
0.  21  666 
0.  21  637 
0. 21  609 
0.  21  581 


0.21  552 
0.  21  524 
0.  21  495 
0.  21  467 
0.  21  438 


0.21  410 
0.21  382 
0.21  353 
0. 21  325 
0.  21  -296 


0.  21  268 
0. 21  240 
0.21  211 
0.21  183 
0.21  155 


0.21  126 
0.21  098 
0.  21  070 
0.  21  041 
0.21  013 


0.  20  985 
0. 20  957 
0.  20  928 
0.20  900 
0.  20  873 


0.  20  844 
0. 20  815 
0.  20  787 
0.20  759 
0.  20  731 


0.20  703 
0. 20  674 
0.  20  646 
0. 20  618 
0. 20  590 


0. 20  562 
0. 20  534 
0.20  505 
0.  20  477 
0. 20  449 
0.  20  421 


9. 93  307 
9.93  299 
9. 93  291 
9.  93  284 
9.93  276 


9.  93  230 
9. 93  223 
9.93  215 
9.  93  207 
9.  93  200 


9.93  192 
9.  93  184 
9.93  177 
9.93  169 
9,  93  161 


9.93  154 
9.93  146 
9.93  138 
9.93  131 
9.  93  123 


9.  93  115 
9.93  108 
9.93  100 
9.  93  092 
9.93  084 
9.  93  077' 
9.  93  069 
9.93  061 
9.93  053 
9.  93  046 


9.93  038 
9.93  030 
9. 93  022 
9.93  014 
9.  93  007 


9.  92  999 
9.  92  991 
9.  92  983 
9.  92  976 
9.  93  968 


9.92  960 
9.  92  953 
9.  92  944 
9.  92  936 

9. 93  929 


9.92  921 
9. 92  913 
9. 92  905 
9.  92  897 
9.  92  889 


9.  92  881 
9. 92  874 
9. 92  866 
9.  92  858 
9. 92  850 
9. 92  842 


29 

1 

0,5 

2 

1,0 

3 

1,4 

4 

1,9 

5 

2,4 

6 

2,9 

7 

3,4 

8 

3,9 

9 

4,4 

10 

4,8 

20 

9,7 

30 

14,5 

40 

19,3 

50 

24,2 

1,4 
1,9 
2,3 
2,8 
3,3 
3,7 
4,2 
4,7 
9,3 
14,0 
18,7 
23,3 


10,5 

10,0 

14,0 

13,3 

17,5 

16,7 

S 

J 

0,1 

0,1 

0,3 

0,2 

0,4 

0,4 

0,5 

0,5 

»;i 

0,6 

0,8 

0,7 

0,9 

0,8 

1,1 

0,9 

1,2 

1,0 

1,3 

1,2 

2,/ 

2,3 

4,0 

3,5 

b,3 

4,7 

6,7 

5,8 

S 

8 

30 

29 

1,9 

1,8 

5,6 

5,4 

9,4 

9,1 

13,1 

12,7 

16,9 

16,3 

20,6 

19,9 

24,4 

23,6 

28,1 

27,2 

]2,2 
15,8 
19,2 
22,8 
26,2 


58° 


286 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Table  XXXVl.—Loganthmio  sines,  cosines,  tantfents,  and  cotaugenis—Contmauii.. 
[Extracted  from  Gauss'  Logaritlimic  and  Trigonometric  Tables.] 

32° 


29 

•2S 

0,5 

0,5 

1,1) 

0,0 

1,4 

1,4 

1,9 

1,0 

2,4 

2,3 

2,9 

2,8 

3,4 

3,3 

3,9 

3,7 

4,4 

4,2 

4,8 

4,7 

9,7 

9,3 

14,5 

14,0 

19,3 

18,7 

■24,2 

23,3 

21 

20 

0,4 

0,3 

0,7 

0,7 

1,0 

1,0 

1,4 

1,3 

1,8 

1,7 

2,1 

2,0 

2,4 

2,3 

2,8 

2,7 

3,2 

3,0 

3,5 

3,3 

7,0 

6,7 

10,5 

10,0 

14,0 

13,3 

17,5 

16,7 

8 

8 

29 

28 

1,8 

1,8 

.5,4 

5,2 

9,1 

8,8 

12,7 

12,2 

10,3 

15,8 

19,0 

19,2 

23,6 

22,8 

27,2 

26,2 

0,0 
1,4 
1,8 
2,2 
2,7 
3,2 
3,6 
4,0 
4,5 
9,0 
13,5 
18,0 
22,5 


0,5 
0,6 
0,7 
0,8 
0,9 
1,0 
1,2 
2,3 
3,5 
4,7 
5,8 


2,0 
6,0 
10,0 
14,0 
18,0 
22,0 
26,0 


S7= 


LOGARITHMS  OF  CIEOULAE  FUNCTIONS. 


287 


Table  XXXVI. — Logariilimic  sin&s,  cosives,  tangents,  and  cotangents — Coutinued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

33^ 


9.73  901 
9.73  921 
9.73  940 
9.73  959 
9.73  978 


9.  73  997 
9.  74  017 
9.  74  036 
9.74  055 
9.  74  074 


9.74  093 
9.74  113 
9.  74  132 
9.74  151 
9.74  170 


9.  74  189 
a.  74  208 
9.74  227 
9.74  246 
9.74  265 


9.74  284 
9.74  303 
9.  74  322 
9.74  341 
9.74  360 


9.74  379 
9.  74  398 
9.  74  417 
9.74  436 
9.  74  455 


9.  74  474 
9.  74  493 
9.74  512 
9.  74  531 
9.  74  549 


9. 74  568 
9.  74  587 
9.74  606 
9. 74  625 
9. 74  644 


9.  74  662 
9.  74  681 
9.  74  700 
9.74  719 
9.74  737 
9.  74  756 


81  252 
81  279 
81  307 
81  335 
SI  362 


81  39U 
81  418 
81  445 
81  473 
81  500j 
81  528 
81  556 
81  583  ! 
81  611  I 
81  638  i 


81  666 
81  693 
81  721 
81  748 
81  776 


81  941 
81  968 

81  996 

82  023 
82  051 


82  078 
82  106 
82  133 
82  161 
82  188 
82  21.5 
82  243 
82  270 
82  298, 
82  325 
82  352 
82  380 
82  407 
82  435 
82  462 


82  544 
82  571 
82  599 


82  626 
82  653 
82  681 


82  762 
82  790 
82  817 


L.  Cotg.   a.  c, 


0. 18  74S 
0. 18  721 
0. 18  693 
0. 18  665 
0.18  638 


0. 18  010 
0. 18  582 
0. 18  .555 
0. 18  527 
0. 18  5110 


0. 18  472 
0. 18  441 
0.  18  417 
0. 18  389 
0. 18  362 


0. 18  334 
0.18  307 
0.  18  279 
0. 18  252 
0.18  224 


0. 18  197 
0.18  169 
0. 18  142 
0.18  114 

0.18  087 


0. 18  059 
0. 18  032 
0. 18  004 
0. 17  977 
0. 17  949 
0^  17  922 
0.17  894 
0. 17  867 
0. 17  839 
0.17  812 
0. 17  785 
0. 17  757 
0. 17  730 
0.17  702 
0.17  675 
0. 17  648 
0.17  620 
0. 17  593 
0.17  565 
0.17  538 


0. 17  511 
0. 17  483 
0. 17  456 
0. 17  429 
0. 17  401 


9. 92  359 
9.92  351 
9.02  343 
9.  92  335 

_9^92  326 
9.  92  3i8" 

.  9.92  310 
9.  92  .302 
9.92  293 
9.92  285 


9.92  277 
9.92  269 
9.92  260 
9.  92  252 
9.  92  24! 


9.92  111 
9.  92  102 
9.  92  094 
9.  92  086 
9.  92  077 
9.  92  069 
9.  92  060 
9.  92  052 
9.  92  044 
9.  92  035 


9.  92  027 
9.  92  018 
9.  92  010 
9.  92  002 
0.  91  993 


9.91  985 
9.91  976 
9.91  968 
9.91  959 
9. 91  951 


9.  91  942 
9.  91  934 
9.91  925 
9.91  917 
9.  91  908 
9.91  900 
9.  91  891  I 
9.91  883 
9.91  874 
9.91  866 
9. 91  857 


14,0 
18,7 
23,3 


0,4 
0,9 
1,4 
1,8. 
2,2 
2,7 
3,2 
3,6 
4,0 
4,5 
9,0 
13,5 
18,0 
22,5 


20 

19 

0'3 

0,3 

0,7 

06 

1,0 

1,0 

1,3 

1,3 

1,7 

1,6 

2,0 

1,9 

2,3 

2,2 

2,7 

2,5 

3,0 

2,8 

3,3 

3,2 

6,7 

6,3 

10,0 

9,5 

13,3 

12,7 

16,7 

15,8 

3,0 
6,0 
9,0 
12,0 
15,0 


9 

9 

2S 

27 

1,6 

1,5  j 

4,7 

4,5 

7,8 

7,5 

10,9 

10,5 

14,0 

13,5 

17,1 

16,5 

20,2 

19,5 

23,3 

-  22,5 

26,4 

25,5 

15,2 
18,6 
21,9 
25,3 


5G'' 


288 


A  MANUAL  OF  TOrOGEAPHIO  METHODS. 


Table  XXXVI. — LogarUlimic  sines,  cosines,  tantjents,  and  cotangents— ContinnaA, 
[Extractcil  from  Gauss'  Logaritlimic  aud  Trigouometric  Tables.] 

34° 


9.  74  756 
9.  74  775 
9.  74  794 
9.74  812 
9.74  831 


9.74  850 
9.  74  863 
9.74  887 
9.74  906 
9.74  024 


9.  74  943 
9.74  901 
9.74  9S0 

9.74  999 

9.75  017 


9.75  036 
9.75  054 
9.  75  073 
9.75  091 
9.  75  110 


9.75  V>i 
9.75  147 
9.75  165 
9.75  184 
9.75  202 


9.  75  221 
9.75  239 
9. 75  258 
9.  75  276 
9.75  294 


9.75  313 
9.75  331 
9,75  350 
9.75  368 
9. 75  386 


9.75  405 
9.  75  423 
9, 75  441 
9.75  459 
9.75  478 
9.75  496 
9.75  514 
9.  75  533 
9.75  551 
9.75  569 
9.75  .587 
9.  75  605 
9.75  624 
9.75  64  2 
9.75  660 
9.75  678 
9.75  696 
9.  75  714 
9.75  733 
9.  75  751 
9.75  769 
9.75  787 
9.  75  805 
9.  75  823 
9.75  841 
9.75  859 


L.  Tang.  il.  o, 


9. 82  899 
9. 82  926 
9.82  953 

9.82  980 

9.83  008 


9.83  171 
9.83  198 
9.  83  225 
9. 83  252 
9.83  280 


9.83  307 
9.83  334 


9.  83  415 


9.  83  442 
9.  83  470 
9.  83  497 
9.83  524 
9.  83  551 


9.  84  119 
9.84  146 
9.84  173 
9.84  200 
9.  84  227 
9.  84  254 
9.84  280 
9.  84  307 
9.  84  334 
9.  84  361 
9.  84  388 
9. 84  415 
9. 84  442 
9.84  469 
9. 84  496 
9.  84  523 


L.  Cotg.   d.  I 


0. 17  047 
0. 17  020 
0.16  992 


0. 16  965 
0. 16  938 
0. 16  911 
0. 10  883 
0. 16  856 


0. 16  829 
0.16  822 
0.10  775 
0. 10  748 
0. 16  720 


0,  16  093 
0. 16  666 
0. 16  639 


9.91  857 
9.91  849 
9.91  840 
9,91  832 
9.91  823 


9.91  815 
9,91  806 
9.91  798 
9,91  789 
9,91  781 


9,91  7' 
9.91  7 
9.91  7: 
9,91  7 
9,91  7 


0. 16  558 
0. 16  530 
0.16  503 
0, 16  476 
0, 16  449 


0, 16  422 
0. 16  395 
0, 16  368 
0. 16  341 
0. 16  314 


0. 16  287 
0. 16  260 
0. 16  232 
0, 16  205 
0. 16  178 


0,16  151 
0, 16  124 
0.16  097 
0.16  070 
0. 10  043 


0.10  016 
0, 15  989 
0, 15  902 
0. 15  935 
0, 15  908 
0, 15  881 
0. 15  854 
0, 15  827 
0. 15  800 
0. 15  773 
0. 15  746 
0. 15  720 
0. 15  693 
0, 15  666 
0. 15  639 
0.15  612 
0, 15  585 
0. 15  5.58 
0.15  531 
0. 15  504 
0. 15  477 


9.91  729 
9.91  720 
9.91  712 
9.91  703 
9,91  695 


9.91  686 
9,91  677 
9,91  069 
9.91  660 
9,91  651 


9,91  643 
9,91  634 
9.91  625 
9.91  017 
9,91  608 


9,91  599 
9,91  .591 
9.91  582 
9.91  ,573 
9.91  565 


9,91  556 
9,91  547 
9.91  538 
9.91  530 
9.91  521 


9.91  512 
9,91  504 
9.91  495 
9.91  486 
9,91  477 
9.91  409 
9.91  460 
9.  91  451 
9.91  442 
9,91  433 
9,91  425 
9,91  416 
9,91  407 

9!  91  389 
9,91  381 
9.91  372 
9.91  363 
9.91  354 
9,91  345 
9,  91  336 


28 
0,5 

27 

0,4 

0,9 

0,9 

1,4 

1,4 

1,9 

1,8 

2,3 

2,2 

2,8 

2,7 

3,3 

3,2 

3,7 

3,6 

4,2 

4,0 

4,7 

4,5 

9,3 

9,0 

14,0 

13,5 

18,7 

18,0 

23,3 

22,5 

1,6 
4,7 
7,8 
10,9 
14,0 
17,1 
20,2 
23,3 
26,4 


0,4 
0,9 
1,3 
1,7 
2,2 
2,6 
3,0 
3,5- 
3,9 
4,3 
8,7 
13.0 
17,3 
21,7 


18 
0,3 
0,6 
0,9 
1,2 
1,5 
1,8 
2,1 
2,4 
2,7 
3,0 
6,0 
9,0 
12,0 
15,0 


0,2 

0,1 

0,3 

0,3 

0,4 

0,4 

(1,6 

0,5 

0,8 

0,7 

0,9 

0,8 

1,0 

0,9 

1,2 

1,1 

1,4 

1,2 

1,5 

1,3 

3,0 

2,7 

4,5 

4,0 

6,0 

5,3 

12,2 
15,8 
19,2 
22,8 
26,2 


8,4 
11,8 
15,2 
18,6 
21,9 
25,3 


55^ 


LOGAEITHMS  OP  CIRCULAE  FUNCTIONS. 


289 


Table  XXXVI. — LogarWwiie  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauas'  Logarithmic  and  Trigonometric  Tables.] 

35° 


9. 75  859 
9.  75  877 
9.  75  895 
9.75  913 
9. 75  931 


9.  75  949 
9.75  967 
9. 75  985 
9.  76  003 
9.  76  021 


9.76  039 
9.  76  057 
9.  76  075 
9.70  093 
9.  76  111 


9. 76  129 
9.  76  146 
9.76  164 
9.76  182 
9.  76  200 


9.  76  218 
9.  76  236 
9.  76  253 
9.  76  271 
9.  76  289 


9.  76  307 
9. 76  324 
9.  76  342 
9.  76  360 
9.76  378 


9. 76  395 
9.  76  413 
9. 76  431 
9.  76  448 
9. 76  466 


9.  76  484 
9.  76  501 
9.  76  519 
9.  76  537 
9.  76  554 


9.  76  572 
9. 76  590 
9.  76  607 
9.  76  625 
9: 76  642 


9.  76  660 
9.  76  677 
9. 76  695 
9. 76  712 
9.  76  730 


9.  76  747 
9. 76  765 
9.76  782 
9.76  800 
9. 76  817 


9.76  835 
9. 76  852 
9.76  870 
9.76  887 
9. 76  904 


9. 76  922 


L.  Tang.   d.  c. 


84  523 
84  550 
84  576 
84  603 
84  630 


84  657 
84  684 
84  711 
84  738 
84  764 


84  791 
84  818 
84  845 
81  872 
84  899 


84  935 
84  952 

84  979 

85  006 
85  033 


85  140 
85  166 


85  193 
85  220 
85  247 
85  273 
85  300 


85  487 
85  514 
85  540 
85  567 


85  594 
85  620 
85  647 
85  674 
85  700 


85  727 
85  754 
85  780 
85  807 
85  834 


85  860 
85  887 
85  913 
85  940 
85  967 


85  993 

86  020 
86  046 
86  073 
86  100 
86  126 


0. 15  477 
0. 15  450 
0. 15  424 
0. 15  397 
0. 15  370 


0. ]5  343 
0.15  316 
0. 15  289 
0. 15  262 
0. 15  236 


0. 15  209 
0. 15  182 
0. 15  155 
0. 15  128 
0. 15  101 


0. 15  075 
0. 15  048 
0. 15  021 
0. 14  994 
0.  14  967 


0. 14  941 
0. 14  914 
0. 14  887 
0. 14  860 
0.  14  834 


0. 14  807 
0. 14  780 
0. 14  753 
0. 14  727 
0.14  700 


0. 14  673 
0. 14  646 
0. 14  620 
0.14  593 
0.  14  566 


0.14  540 
0.14  513 
0. 14  486 
0.  14  460 
0. 14  433 


0. 14  406 
0. 14  380 
0. 14  353 
0. 14  326 
0. 14  300 


0. 14  273 
0. 14  246 
0. 14  220. 
0. 14  193 
0. 14  166 


9.  91  336 
9.91  328 
9.91  319 
9. 91  310 
9. 91  301 


9. 91  292 
9.91  283 
9.91  274 
9.91  266 
■9.91  257 


9.  91  248 
9.  91  239 
9.91  230 
9.91  221 
9.91  212 


9. 91  203 
9.91  194 
9.91  185 
9.91  176 
9.91  167 


9.91  158 
9.91  149 
9.91  141 
9.91  132 
9.91  123 


9.91  114 
9.91  105 
9.91  096 
9.91  087 
9.91  078 


9.91  069 
9.91  060 
9.91  051 
9. 91  042 
9.91  033 


9. 91  023 
9.91  014 
9.91  005 
9. 90  996 
9.90  987 


9.90  978 
9. 90  969 
9. 90  960 
9.90  951 
9.90  942 


0.  14  140 
0. 14  113 
0. 14  087 
0. 14  060 
0. 14  033 


0. 14  007 
0.  13  980 
0. 13  964 
0. 13  927 
0. 13  900 


0. 13  874 


L.  Cotg.   d.  c.   L.  Tang.    L.  Siu.   d 


9.90  887 
9.90  878 
9.90  869 


9.  90  823 
9.  90  814 
9.  90  805 


27 

2« 

0,4. 

0,4 

0,9 

0,9 

1,* 

1,3 

1,8 

1,7 

2,2 

2,2 

2,7 

2,6 

3,2 

3,0 

3,6 

3,5 

4,0 

3,9 

4,5 

4,3 

9,0 

8,7 

13,5 

13,0 

18,0 

17,3 

22,5 

21,7 

17 

10 

9 

0,3 

0,2 

0,2 

0,6 

0,3 

0,3 

0,8 

0,5 

0,4 

1,1 

0,7 

0,6 

1,4 

0,8 

0,8 

1,7 

1,0 

0,9 

2,0 

1,2 

1,0 

2,3 

1,3 

1,2 

2,6 

1,5 

1,4 

2,8 

1,7 

1,5 

5,7 

3,3 

3,0 

8,5 

.5,0 

4,5 

11,3 

e,7 

0,0 

14,2 

8,3 

7,5 

9,0 
12,0 
15,0 


-19 


54° 


290 


A  MANUAL  OP  TOPOGEAPHIC  METHODS. 


Table  XXXVI. — Logarillimic  sines,  cosines,  tangents,  and' cotiingenis — Continued. 
[Extracted  from  Gauss'  Logaritlimic  anil  Trigonometric  Tables.] 

36° 


9.76  922 
9.76  939 
9.76  957 
9.  76  974 
9.  76  991 


9.77  009 
9. 77  026 
9. 77  043 
9. 77  061 
9.  77  078 


9. 

9.77  164 


095 


147 


9.77  181 
9.77  199 
9. 77  216 
9.  77  233 
9.  77  250 


9.  77  268 
9.77  285 
9.77  302 
9.77  319 
9.  77  336 


9.  77  353 
9.77  370 
9.77  387 
9.  77  405 
9.  77  432 


9.  77  456 
9.  77  473 
9. 77  490 
9.77  507 


9. 77  524 
9. 77  541 
9.  77  558 
9.  77  575 
9.  77  592 


9.77  609 
9.77  626 
9.  77  G43 
9.  77  660 
9.  77  677 


9.  77  694 
9.  77  711 
9.77  728 
9.  77  744 
£.  77  761 


9.77  778 
9.77  795 
9.77  812 
9.77  829 
9.77  846 


L.  Tang.   d.  c 


9.  86  126 
9.  86  153 
9.86  179 
9. 86  206 
9.  86  232 


9. 86  269 
9.  86  285 
9.86  312 
9. 86  338 
9. 86  365 


9. 86  392 
9.  86  418 
9.  86  445 
9. 86  471 
9.  86  498 


9.86  656 
9.  86  683 
9.  86  709 
9.  86  736 


9.  86  815 
9.  86  842 
9.  86  868 
9.  86  894 


9.  86  921 
9.  86  947 
9.  86  974 
9.87  000 
9.  87  027 


9.  87  053 
9. 87  079 
9.87  106 
9. 87  132 
9.87  158 


9.87  185 
9.  87  211 
9.  87  238 
9.87  264 
9.  87  290 


9.87  317 
9. 87  343 
9.  87  369 
9.  87  396 
9.  87  422 


9.87  448 
9. 87  475 
9.87  501 
9.87  527 
9. 87  554 


9.87  580 
9.87  606 
9.87  633 
9.87  659 
9.87  685  I 
9.  87  711  I 


L.  Cotg.  '  d. 


L.  Cotg. 


0. 13  874 
0. 13  847 
0. 13  821 
0. 13  794 
0. 13  768 


0. 13  741 

0. 13  715 

0. 13  688 

0. 13  662 

0. 13  635 


0.  J  3  608 

0. 13  582 

0. 13  555 

0. 13  529 

0. 13  502 


0. 13  476 
0. 13  449 
0. 13  423 
0. 13  397 
0.13  370 


0. 13  344 
0.13  317 
0.  13  291 
0. 13  264 
0. 13  238 


0.  13  211 
0.  13  185 
0.13  158 
0.13  132 
0. 13  106 


0. 13  079 
0. 13  053 
0. 13  026 
0.13  000 
0. 12  973 


0. 12  947 
0. 12  921 
0. 12  894 
0. 12  868 
0. 12  842 


0. 12  815 
0. 12  789 
0. 12  762 
0. 12  736 
0.12  710 


0. 12  683 
0. 12  657 
0. 12  631 
0.12  604 
0.12  578 


0. 12  552 
0. 12  525 
0. 12  499 
0. 12  473 
0. 12  446 


0. 12  420 
0. 12  394 
0.12  367 
0. 12  341 
0. 12  315 
0. 12  289 


9.  90  796 
9. 90  787 
9.  90  777 
9.  90  768 
9.  90  759 


9.90  750 
9.  90  741 
9.90  731 
9. 90  722 
9.90  713 


9.  90  657 
9.90  648 
9. 90  639 
9.90  630 
9.90  620 


9.90  611 
9.90  602 
9.  90  692 


9.  90  565 
9.  90  555 
9. 90  546 


9.90  424 
9.  90  415 
9.  90  406 
9.  90  396 


9.  90  311 
9.90  301 
9.90  292 


9.90  263 
9.  90  264 
9.  90  244 


0,9 
1/4 
1,8 
2,2 
2,7 
3,2 
3,6 
4,0 
4,5 
9,0 
13,5 
18'0 
22,5 


0,4 
0,9 
1,3 
1,7 
2,2 
2,6 


8,7 
13,0 
17,3 
21,7 


18 

17 

0,3 

0,3 

0,6 

0,6 

0,9 

0,8 

1,2 

1/1 

1,5 

M 

1,8 

1/7 

2,1 

2,0 

2,4 

2,3 

2,7 

2,6 

3,0 

2,8 

6,0 

5,7 

9,0 

8'5 

12,0 

11,3 

15,0 

14,2 

0,4 
0,6 
0,8 
0,9 
1,0 
1,2 
1/4 
1,5 
3,0 
4,5 


4,5 

4'3 

V,b 

7,2 

l(),.'i 

10,1 

13,5 

13,0 

I6„S 

15,9 

19,6 

18,8 

22,5 

21,7 

2b,b 

24,6 

53° 


LOGARITHMS  OF  CIECULAE  FUNCTI02fS. 


291 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extraoted  from  Ganss'  Logarithmic  and  Trigonometric  Tables.] 

37° 


9.  77  946 
9.  77  963 
9.77  980 
9.77  997 
9.  78  013 


9.  78  030 
9.  78  047 
9.  78  063 


9.  78  130 
9.78  147 
9.  78  163 
9.  78  180 


9.  78  197 
9.  78  213 
9.  78  230 
9.  78  246 
9.  78  263 
9.  78  280 
9.  78  296 
9.  78  313 
9.  78  329 
9.  78  346 


9.  78  395 
9. 78  412 
9.  78  428 


9. 78  445 
9.  78  461 
9. 78  478 
9. 78  494 
9.  78  510 


9.  78  527 
9.78  543 
9. 78  560 
9. 78  576 
9.  78  592 


9.  78  609 
9.  78  625 
9.  78  642 
9. 78  658 
9. 78  674 


9.  78  691 
9.  78  707 
9.  78  723 
9.  78  739 
9.  78  756 


9.78  772 
9.  78  788 
9.  78  805 
9. 78  821 
9.  78  837 


L.  Tang,   d 


87  764 
87  790 
87  817 


87  843 
87  869 
87  895 
87  922 


;  071 


88  105 
88  131 
88  158 
88  184 
88  210 


88  629 
88  655 
88  681 


89  125 


89  151 
89  177 
89  203 
89  229 
89  255 
89  281 


L.  Cotg.   d.  c. 


0.12  289 
0. 12  262 
0. 12  236 


0. 12  105 
0. 12  078 
0. 12  052 


0. 12  026 
0. 12  000 
0. 11  973 
0.11  947 
0. 11  921 


0. 11  895 
0. 11  869 
0. 11  842 
0.11  816 
0. 11  790 


0. 11  764 
0. 11  738 
0. 11  711 
0.11  685 
0. 11  659 


0.11  633 
0. 11  007 
0. 11  580 
0. 11  554 
0.11  528 


0. 11  502 
0. 11  476 
0. 11  450 
0. 11  423 
0. 11  397 


0. 11  371 
0. 11  345 
0. 11  319 
0. 11  293 
0. 11  267 


0. 11  241 
0. 11  214 
0.11  188 
0. 11  162 
0. 11  136 


0.11  110 
0. 11  084 
0. 11  058 
0. 11  032 
0.11  006 


0. 10  980 
0. 10  954 
0. 10  927 
0. 10  901 
0. 10  875 


0. 10  849 
0. 10  823 
0. 10  797 
0.10  771 
0. 10  745 
0. 10  719 


L.  Tang. 


90  235 
90  225 
90  216 
90  206 
90  197 


90  187 
90  178 
90  168 
90  159 
90  149 


90  120 
90  111 
90  101 


89  947 
89  937 
89  927 
89  918 


89  702 
89  693 
89  683 
89  673 


0,4 

0,4 

0,9 

0,9 

1,4 

1,3 

1,« 

1,7 

2,2 

2,2 

2,7 

2,6 

3,2 

3,0 

3,6 

3,5 

4,0 

3'9 

4,6 

4,3 

9,0 

8,7 

13,5 

13,0 

18,0 

17,3 

22,5 

21,7 

17 

16 

0,3 

0,3 

0,6 

0,5 

0,8 

0,8 

1,1 

1,1 

l,i 

1,3 

I'V 

1,6 

2'() 

1,9 

2,3 

2,1 

2'6 

2,4 

2,8 

2,7 

5,7 

5,3 

K,5 

8,0 

11,3 

10,7 

14,2 

13,3 

10 

9 

0,2 

0,2 

0,3 

0,3 

0,5 

0,4 

0,7 

0,6 

0,8 

0,8 

1,0 

0,9 

1,2 

1,0 

1,3 

1,2 

1,5 

1,4 

1,7 

1,5 

3,3 

3,0 

5,0 

4,5 

6,V 

6,0 

8,3 

7,5 

1,4 

1,3 

4,1 

3,9 

6,8 

6,5 

9,4 

9,1 

12,2 

11,7 

14,8 

14,3 

17,6 

16,9 

20,2 

19,5 

22,9 

22,1 

2b,6 

24,7 

52= 


292 


A  IMANUAL  OF  TOPOGKAPlilO  METUODS. 


Table  XXXVI. — Loiiayillimic 
[Extracted  from  Gai 


/«(«,  cosines,  laiigents,  and  cotangents — Coutinued. 
.s'  Lojiarithmic  aud  Trigonometric  Tables.] 

38° 


«50 
967 

9. 78  nS3 
_9.7S_0!)9 

9.79  015" 
9.  79  031 
9.  79  047 
9.  79  063 
9.  79  079 

"977Dl)9T 
9.  79  111 
9.  79  128 
9.  79  144 
9.  79  160 
9.  79  176 
9.  79  192 
9.  79  208 
9.  79  224 
9. 79  240 


■9  256 


9.  79  383 
9^9^399 
9.  79  415 
9.  79  431 
9.  79  447 
9.  79  463 
9.  79  478 
9. 79  494 
9.  79  510 
9.79  526 
9.  79  542 
9.79  558 


9.  79  573 
9.  79  589 
9.  79  605 
9. 79  621 
9.  79  636 


9.79  652 
9.  79  668 
9. 79  684 
9.79  699 
9.79  715 


9.79  731 
9.  79  746 
9.79  762 
9.  79  778 


9. 79  825 
9.79  840 
9.  79  856 
9. 79  872 


L.  Tang.   d.  c. 


9. 89  801 
9.89  827 
9.89  853 


9.90  112 
9.  90  138 
9.90  164 


9.  90  190 
9.  90  216 
9.  90  242 
9.  90  268 
9.  90  294 


.9.90  ; 


346 


9. 90  449 
9. 90  475 
9.  90  501 
9.  90  527 
9. 90  553 


9.90  578 
9.90  604 
9.90  630 
9. 90  656 
9.90  682 
9.  90  708 
9. 90  734 
9.90  759 


L.  Cotg. 


0. 10  719 
0. 10  693 
0.  10  667 
0. 10  641 
0^10_615 
0. 10  589 
0. 10  563 
0.  10  537 
O 10  511 
0. 10  485 


0. 10  459 
0.10  4« 
0. 10  407 
0. 10  381 
0. 10  355 
0. 10  329 
0. 10  303 
0.10  277 
0. 10  251 
0. 10  225 


0.  10  199 
0. 10  173 
0.10  147 
0. 10  121 
0. 10  095 


0. 10  069 
0.10  043 
0.10  017 
0. 09  99] 
0.  09  965 


0.09  939 
0.  09  914 
0.09  888 
0.  09  862 
0.  09  836 


0. 09  810 
0.  09  784 
0.09  758 
0.  09  732 
0.  09  706 


0.  09  551 
0. 09  525 
0. 09  499 
0.  09  473 
0.  09  447 


0.  09  422 
0.  09  396 
0.09  370 
0.  09  344 
0.09  318 
0.09  292 
0.09  266 
0.  09  241 
0.  09  215 
0.09  189 


L.  Cos. 


9. 89  633 

9.89  624 

J)JS9  61£ 
9.  89' 004 

9.89  594 

9.  89  584 

9.  89  574 

9.  89  564 


9.89  524 
9.  89  514 
9.89  504 
9.89  495 
9.  89  485 
9.  89  475 
J3^89  465 
9.  .89  455" 
9.  89  445 
9.  89  435 
9.  89  425 
9.  89  415 


9.89  405 
9.  H9  395 
9.  89  385 
9.  89  375 
9.  89  364 


9.  89  354 
9.  89  344 
9.  89  334 
9.  89  324 
9.  89  314 


9.  89  304 
9.  89  294 
9.89  284 
9.89  274 
9.89  264 
9:89  254 
9.89  244 
9.89  233 


9.  89  152 
9.89  142 
9.89  132 
9.89  122 


9.  89  091 
9.89  081 
9.89  071 
9.89  060 


P.P. 


4,3 
8,7 
13,0 
17,3 
21,7 


0,4 
0,8 
1,2 
1,7 
2,1 
2,5 
2,9 
3,3 
3,8 
4,2 
8,3 
12,5 
16,7 
20,8 


17 

0,3 

16 

0,3 

0,6 

0,5 

0,8 

0,8 

1,1 

1,1 

1,4 

1,3 

1/7 

1,6 

2,0 

1,9 

2,3 

2,1 

2,6 

2,4 

2,8 

2,7 

5,7 

.5,3 

8,5 

8,0 

11,3 

10,7 

14,2 

13,3 

11 

10 

0,2 

0,2 

0,4 

0,3 

0,6 

0,5 

0,7 

0,7 

0,9 

0,8 

1/1 

1,0 

1,3 

1,2 

1,5 

1,3 

1,6 

1,5 

1,8 

1,7 

3,7 

3,3 

5,5 

5,0 

7,3 

6,7 

9,2 

8,3 

10 

10 

26 

25 

1,3 

1/2 

3,9 

3,8 

6,5 

6,2 

9,1 

8,8 

11,7 

11,2 

14,3 

13,8 

16,9 

16,2 

19,5 

18,8 

22,1 

21,2 

24,7 

23,8 

1,2 
1,5 
1,8 
2,0 
2,2 
2,5 
5,0 
7,5 
10,0 
12,5 


1/4 
4,3 
7,2 
10,1 
13,0 
15,9 
18,8 
21,7 
24,6 


31' 


LOGAEITHMS  OF  CIRCULAR  FUNCTIONS. 


293 


Table  XXXVI. — Logarithmic  s'mes,  cosines,  tangents^  and  cotangents — Continued, 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

39° 


9. 79  903 
9.79  918 
9.  79  934 
9.79  950 


9.79  905 
9.79  981 

9.79  990 

9.80  012 
9  80  027 


9.80  043 
9.80  058 
9. 80  074 
9.  80  C89 
9.80  105 


9.80  120 
9.80  136 
9.80  151 
9.  80  160 
9.80  182 


9.80  197 
9.  80  213 


9.  80  274 
9.  80  290 
9.80  305 
9. 80  320 


9.80  428 
9. 80  443 
9.  80  458 
9. 80  473 
9.  80  489 


9.80  504 
9. 80  519 
9. 80  534 
9. 80  550 
9.80  565 


9.80  580 
9.  80  595 
9. 80  610 
9.  80  625 
9.  80  641 


9.  80  656 
9. 80  671 
9.  80  686 
9. 80  701 
9.  80  716 


90  837 
90  863 
90  889 
90  914 
90  940 


90  906 

90  992 

91  018 
91  043 
91  009 


91  095 
91  121 
91  147 
91  172 
91  198 


91  224 
91  250 
91  276 
91  301 
91  327 


91  353 
91  379 
91  404 
91  430 
91  456 
91  482 
91  507 
91  533 
91  559 
91  585 


91  610 
91  636 
91  662 
91  688 
91  713 


91  739 
91  765 
91  791 
91  816 
91  842 


91  868 
91  893 
91  919 
91  945 
91  971 


91  996 

92  022 
92  048 
92  073 
92  099 


92  125 
92  150 
92  176 
92  202 
92  227 


92  253 
92  279 
92  304 
92  330 
92  356 
92  381 


0.  09  163 
0.09  137 
0.09  111 
0.  09  086 


0.08  879 
0.  08  853 
0.  08  828 
0.08  802 


0. 08  750 
0.  08  724 
0.  08  099 
0.  08  673 


0.  08  617 
0.  08  621 
0.  08  596 
0.  08  570 
0.  08  544 


0.  08  518 


0.1 


415 


0.08  390 
0.08  364 
0.08  338 
0.  08  312 
0.  08  287 


0.  08  261 
0.  08  235 
0.  08  209 
0.08  184 
0.08  158 


0.  08  132 
0. 08  107 
0.  08  081 
0. 08  055 
0.  08  029 


0.  08  004 
0.  07  978 
0.  07  952 
0. 07  927 
0.  07  901 


0. 07  875 
0.  07  850 
O:  07  824 
0. 07  798 
0.07  773 


0.  07  747 
0.  07  721 
0. 07  696 
0. 07  670 
0. 07  644 
0.07  619 


L.  Tang. 


88  793 
88  782 
88  772 
88  761 


88  678 
88  668 
88  657 
88  647 


88  636 
88  626 
88  615 
88  605 
88  594 


88  573 
88  563 
88  552 
88  542 


n,4 

0,4 

0,9 

0,8 

1,3 

1,2 

1,V 

1,7 

2,2 

2,1 

2,6 

2,,') 

3,0 

2,9 

3,!> 

3,3 

3'9 

3,8 

4,3 

4,2 

8,3 

13,0 

12,5 

ri,3 

16,7 

21,7 

20,8 

1 

16 

0,3 

2 

0,5 

3 

0,8 

4 

1/1 

5 

1,3 

6 

1,6 

V 

1,9 

H 

2,1 

9 

2,4 

10 

2,7 

20 

5,3 

30 

8'0 

40 

10,7 

50 

13'3 

1,2 

1,J 

3,6 

3,4 

b,9 

5,7 

8,3 

7,9 

10,6 

10,2 

13,0 

12,5 

lb,4 

14,8 

1V,V 

17,1 

20,1 

19,3 

22,5 

21,6 

24,8 

23,9 

50= 


294 


A  MANUAL  OF  TOPOGEAPHIC  METHODS. 


Table  XXXVI. — Locjaritiimic  sines,  cosines,  tangents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

40° 


9.80  837 
9. 80  852 
9. 80  867 


9.  80  987 
9.81  002 
9.81  017 


9.  81  032  i 
9.81  047 
'9.81  061 
9.  81  076  , 
9.8]  091 


9.  81  106 
9.81  121 
9.81  136 
9.81  151 
9.81  166 


9.  SI  180 
9.81  195 
9.  81  210 
9.81  225 
9.81  240 


9. 81  254 
9.81  269 
9. 81  284 
9.81  299 
9.81  314 


9.81  328 
9.81  343 
9.81  358 
9.  81  372 
9.  81  387 


9.  81  402 
9.81  417 
9.81  431 
9.  81  446 
9.81  461 


9.  81  475 
9. 81  490 
9.81  505 
9.  81  519 
9. 81  534 


9.  81  549 
9.81  563 
9. 81  578 
9.  81  592 
9.81  607 
9. 81  622 
9.81  636 
9.81  651 
9.81  665 
9.  81  680 
9.  81  694 


9.92  381 
9.  92  407 
9.  92  433 
9.  92  458 
9. 92  484 


9.  92  510 
9.  92  535 
9.  92  561 
9.  92  587 
9. 92  612 


9. 92  638 
9.92  663 
9.  92  689 
9.  92  715 
9. 92  740 


9. 92  766 
9. 92  792 
9.  92  817 
9. 92  843 
9.  92  868 


9.  92  894 
9.  92  920 
9. 92  945 
9.  92  971 
9. 92  996 


9.93  022 
9.  93  048 
9.93  073 
9.93  099 
9.93  124 


9.93  150 
9. 93  175 
9.93  201 
9.  93  227 
9.93  252 


9. 93  406 
9.  93  431 
9.  93  457 
9.  93  482 
9.93  508 


9.93  533 
9.93  559 
9.93  584 
9. 93  610 
9.93  636 


9.93  661 
9. 93  687 
9.93  712 
9.  93  738 
9.93  763 
9.93  789 
9.  93  814 
9.  93  840 


0.  07  619 
0.  07  593 
0.07  667 
0.07  542 
0.07  516 


0.  07  490 
0.07  465 
0.  07  439 
0. 07  413 
0.07  388 


0.U7  362 
0.  07  337 
0.07  311 
0.  07  285 
0.  07  260 


0.  07  234 
0.07  208 
0.07  183 
0.  07  157 
0.07  132 


O.07  106 
0.07  080 
0.  07  055 
0.07  029 
0.07  004 


0. 06  978 
0.06  952 
0.  06  927 
0.06  901 
0.  06  876 


0.06  850 
0.  06  825 
0.  06  799 
0.06  773 
0.  06  748 


0.1 


:  722 


0.06  697 
0.06  671 
0.  06  646 
0.  06  620 


0.  06  594 
0.  06  569 
0.  06  543 
0. 06  518 
0.  06  492 


0.06  467 
0. 06  441 
0. 06  416 
0.  06  390 
0.  06  364 


0.06  339 
0. 06  313 
0.06  288 
0.  06  262 


0.1 


0.06  211 
0.06  186 
0.06  160 
0.06  135 
0.  06  109 
0.  06  084 


d.  c.   L.  Tang. 


88  372 
88  362 
88  351 
88  340 


88  201 
88  191 
88  180 
88  169 


88  105 
88  094 
88  083 


88  040 
88  029 
88  018 
88  007 


87  985 
87  975 
87  964 
87  953 


87  942 
87  931 
87  920 
87  909 
87  898 


87  887 
87  877 
87  866 
87  855 
87  844 
87  833 
87  822 
87  811 


0,4 

0,4 

0,9 

0,8 

1,3 

1,2 

1/' 

i:i 

2,2 

2,1 

2,6 

2,b 

3,0 

2,9 

3,5 

3,3 

3,9 

3,8 

4,3 

4,2 

8,V 

8,3 

4,7 
7,0 
9,3 
11,7 
10 
0,2 
0,3 
0,5 


n 

10  1 

26 

26 

1,2 

1,3 

3,5 

3,9 

5,9 

6,5 

8,3 

9,1 

10,6 

11,7 

13,0 

14,3 

15,4 

16,9 

17,7 

19,5 

20,1 

22,1 

22,5 

24,7 

24,8 

1,2 
3,8 
6,2 
8,8 
11,2 
13,8 
16,2 
18,8 
21,2 
23,8 


49= 


LOGAEITHMS  OF  CIEOULAE  FUNCTIONS. 


295 


Table  XXXVI. — Logarithmic  sines,  cosines,  tangents^  and  cotangents — Continued. 

[Extractetl  from  Gauss'  Logarithmic  and  Trigonometric  Tables.] 

410 


L.  Tang. 


L.  Cotg. 


81  694 
81  709 
81  723 
81  738 
81  752 


81  767 
81  781 
81  796 
81  810 
81  825 


81  9U 
81  926 
81  940 
81  955 
81  969 


81  983 

81  998 

82  012 
82  026 
82  041 


82  055 
82  069 
82  084 
82  098 
82  112 


82  126 
82  141 
82  155 
82  169 
82  184 


82  198 
82  212 
82  226 
82  240 
82  255 


82  269 
82  283 
82  297 
82  311 
82  326 


82  340 
82  354 
82  368 
82  382 


82  410 
82  424 
82  439 
82  453 
82  467 


9. 93  916 
9.  93  942 
9. 93  967 

9. 93  993 

9.94  018 


9.94  044 
9.  94  069 
9.  94  095 
9.  94  120 
9.  94  146 


9.  94  171 
9.  94  197 
9.  94  222 
9.94  248 
9.  94  273 


9.94  299 
9.94  324 
9. 94  350 
9.94  375 
9.  94  401 
9. 94  426 
9.  94  452 
9. 94  477 
9.  94  503 
9.  94  .528 


9. 94  554 
9. 94  579 
9. 94  604 
9.94  630 
9.  94  655 


9.  94  681 
9.  94  706 
9.94  732 
9. 94  757 
9.  94  783 


9. 95  062 
9.  95  088 
9.95  113 
9.  95  139 
9.95  164 


9.95  190 
9.95  215 
9. 95  240 
9.  95  266 
9.  95  291 
9. 95  317 
9.  95  342 
9. 95  368 
9.  95  393 
9. 95  418 
9.95  444 


0.  06  084 
0. 06  058 
0. 06  033 
0.06  007 
0.  05  982 


0.05  956 
0.  05  93] 
0.  05  905 
0.  05  880 
0. 05  854 


0. 05  829 
0. 05  803 
0.05  778 
0.  05  752 
0. 05  727 


0.  05  701 
0.  05  676 
0.05  650 
0.  05  625 
0. 05  599 


0.  05  574 
0.  05  548 
0.  05  523 
0.  05  497 
0.05  472 


0.05  446 
0.  05  421 
0.  05  396 
0.  05  370 
0.  05  345 


0.  05  319 
0. 05  294 
0. 05  268 
0.  05  243 
0.  05  217 


0.05  192 
0.05  166 
0.05  141 
0.05  116 
0.  05  090 


0. 05  065 
0.05  039 
0.05  014 
0. 04  988 
0.  04  963 


0. 04  938 
0.  04  912 
0. 04  887 
0. 04  861 
0.  04  836 


0.04  810 
0.  04  785 
0.04  760 
0.  04  734 
0.  04  709 
0.04  683 
0.  04  658 
0.  04  632 
0.  04  607 
0.04  582 
0. 04  556 


87  778 
87  767 
87  756 
87  745 
87  734 


87  723 
87  712 


87  668 
87  657 
87  646 
87  635 
87  624 


87  613 
87  601 
87  590 


87  546 
87  535 
87  524 
87  513 


87  501 
87  490 
87  479 


87  446 
87  434 
87  423 
87  412 
87  401 


87  390 
87  378 
87  367 
87  356 
87  345 


87  334 
87  322 
87  311 
87  300 

87  288 


9.87  277 
9.87  266 
9.87  255 
9.87  243 
9.87  232 


221 
9. 87  209 


9.87  141 
9.87  130 
9.87  119 


0,4 

0,4 

0,9 

0,8 

1,3 

1,2 

1;7 

1,7 

2,2 

2,1 

2,6 

2,5 

3,0 

2,9 

3,5 

3,3 

3,9 

3,8 

4,3 

4,2 

«,v 

8,3 

13,0 

12,5 

17,3 

16,7 

21,7 

20,8 

15 

14 

0,2 

0,2 

0,5 

0,5 

0,8 

0,7 

1,0 

0,9 

lr2 

1,2 

l,c 

1,4 

1,8 

1,6 

2,0 

1,9 

2,2 

2,1 

2,5 

2,3 

5,0 

4,7 

V,5 

7,0 

10,0 

9,3 

12,b 

11,7 

12 

11 

0,2 

0,2 

0,4 

0,4 

0,6 

0,6 

0,8 

0,7 

1,0 

0,9 

1,2 

1,1 

1,4 

1,3 

1,6 

15 

1,8 

1,6 

2,0 

1,8 

4,0 

3,7 

6,0 

5,5 

8,0 

7,3 

10,0 

9,2 

12 

12 

26 

25 

1,1 

1,1 

8,2 

3,1 

5,4 

5,2 

7,6 

7,3 

9,8. 

9,4 

11,9 

11,5 

14,1 

13,5 

16,2 

15,6 

18,4 

17,7 

20,6 

19,8 

22,8 

21,9 

24,9 

23,9 

1,1 

3,4 
5,7 
7,9 
10,2 
12,5 
14,8 
17,1 
19,3 
21,6 
23,9 


48° 


296 


A  MxiNUAL  OF  TOPOGKAPHIC  METHODS. 


Table  XXXVI. — Loiiarillimic  siiics,  cosines,  tangenis,  and  coteH(/e»/s— Continued. 
[Extraotocl  Iroin  Gausa'  Logarithmic  and  Trigonometric  TivViles.) 

42° 


47c 


LOGAEITHMS  OF  CIEGULAR  FUNCTIONS. 


297 


Table  XXXVI. — Logarithmic  sines,  cosines,  tanffents,  and  cotangents — Continued. 
[Extracted  from  Gauss'  Logarithmic  and  Trigonometric  Tables-] 

430 


83  378 
83  392 
83  405 
83  419 
83  432 


83  441) 


83  513 
83  527 
83  540 


83  661 
83  674 
83  688 
83  701 
83  715 
83  728 
83  741 
83  755 


83  901 
83  914 
83  927 
83  940 
83  954 
83  967 
S3  980 

83  993 

84  006 
84  020 
84  033 
84  046 
84  059 
84  072 
84  085 
84  098 
84  112 
84  125 
84  138 
84  151 
84  164 
84  177 


9. 97  016 
9.  97  042 
9.  97  067 


9. 97  345 
9. 97  371 
9.  97  396 
9. 97  421 

9. 97  447 


9. 97  472 
9.97  497 
9.  97  523 
9.  97  548 
9.97  573 
9.97  698 
9.  97  624 
9.97  649 
9.  97  674 
9.97  700 
9.97  725 
9. 97  750 
9.  97  776 
9.  97  801 
9.97  826 
9. 97  851 
9.  97  877 
9.97  902 
9.97  927 
9.97  953 


9.98  180 
9.98  206 
9.  98  231 
9. 98  256 
9.  98  281 
9.  98  307 
9.  98  332 
9.  98  357 


L.  Cotg.   d.  c. 


0.  03  034 
0. 03  009 
0. 02  984 
0. 02  958 
0.  02  933 


0. 02  908 
0. 02  882 
0.02  857 
0.  02  832 
0.  02  807 


0.  02  781 
0.02  756 
0.  02  731 
0. 02  705 
0.  02  6fi0 


0.02  655 
0.02  629 
0.  02  604* 
0.  02  579 
0.  02  553 


0.  02  528 
0.  02  503 
0. 02  477 
0.  02  452 
0. 02  427 
0.  03  402 
0.  02  376 
0. 02  351 
0.  02  326 
0.02  300 
0.  02  275 
0.02  250 
0,  02  224 
0.02  199 
0.  02  174 
0.02  149 
0.  02  123 
0.  02  098 
0.  02  073 
0.02  047 
0.02  022 
0.  01  997 
0.01  971 
0.01  946 
0.  01  921 
0.01  896 
0.01  870 
0.01  845 
0.01  820 
0.01  794 
0.  01  769 
0.01  744 
■  0.  01  719 
0.01  693 
0.  01  668 
0.  01  643 
0.  01  617 
0.01  592 
0.  01  567 
0. 01  542 
0.01  516 


1  200 


86  176 
86  164 
86  152 
86  140 
86  128 
86  116 
86  104 
86  092 
86  080 
86  068 


85  900 
85  888 
85  876 
85  864 
85  851 
83  839 
85  827 
85  815 
85  803 
85  791 
85  779 
85  766 
85  754 
85  742 
85  730 
85  718 
85  706 
85  693 


9,0 
11,0 
13,0 
10,0. 
17,0 
19,0 
21,0 
23,0 
25,0 


0,4 

0,4 

0,9 

0,8 

1,3 

1,2 

1,Y 

1,7 

2,2 

2,1 

2,6 

2,5 

3,0 

2,9 

3,5 

3,3 

3,9 

3,8 

4,3 

4,2 

8,7 

8,3 

13,0 

12,.T 

l/,3 

16,7 

21,7 

20,8 

14 

IS 

0,2 

0,2 

0,5 

0,4 

0,7 

0,6 

0,9 

0,9 

1,2 

1,1 

1,4 

1,3 

1,6 

1,5 

1,9 

1,7 

2,1 

2,0 

2,3 

2,2 

4,V 

4,3 

7,0 

6,,') 

9,3 

8,V 

11,V 

10,8 

12 

11 

0,2 

0,2 

0,4 

0,4 

(1,6 

0,6 

0,8 

0,7 

1,0 

0,9 

1,2 

i,l 

1,4 

1,3 

1,6 

l,.*) 

1,8 

1,6 

2,0 

1,8 

4,0 

3,V 

6,0 

b,5 

8,0 

7,3 

1U,0 

9,2 

8,7 
10,6 
12,5 
14,4 
16,3 
18,3 
20,2 
22,1 
24,1 


1,1 
3,1 
5,2 
7,3 
9,4 
11,5 
13,5 
15,6 
17,7 
19,8 
21,9 
23,9 


46° 


298 


A  MANUAL  OF  TOPOGRAPHIC  METHODS. 


Ta^le  XXXVI. — Loijarithmic  sines,  cosmesj  tangents,  and  cotangents — Continued. 

[Extracted,  from  Crauss'  LogiU'itlimic  and  Trigonometric  Tables.] 

440 


9.  84  177 
9.  84  190 
9. 84  203 
9.84  216 
9.84  229 


9.  84  308 
9.  84  321 
9.  84  334 
9.  84  347 
9.84  360 


9.  84  373 
9.  84  385 
9.  84  308 
9.  84  411 
9.  84  424 


9. 84  437 
9.  84  450 
9.  84  463 
9.  84  476 
9.  84  489 
9. 84  502 
9.84  515 
9.84  528 
9.  84  540 
9.  84  553 
9.  84  566 
9.84  579 
9. 84  592 
9.84  605 
9.  84  618 
9.  84  630 
9.84  643 
9.  84  656 
9.  84  669 


9.  84  707 
9.  84  720 
9.  84  733 
9.  84  745 
9.84  75S 
9.  84  771 
9. 84  784 
9.  84  796 
9.  84  809 
9.84  822 
9.84  835 
9.84  847 
9. 84  860 
9.84  873 
9. 84  885 
9.  84  898 
9. 84  911 
9.  84  923 
9.84  936 
9.  84  949 


9.98  509 
9.98  534 
9.  98  560 
9.  98  585 


9.  98  610 
9. 98  635 
9.  98  681 
9.98  686 


9.  98  863 
9.  98  888 
9.98  913 
9. 98  939 
9.  98  964 


9.  98  989 
9.99  015 
9.  99  040 
9.99  065 
9.99  090 
9.99  116 
9.99  141 
9.99  166 
9.99  191 
9.99  217 
9.  99  242 
9.99  267 
9.99  293 
9.  99  318 
9.99  343 
9.99  368 
9.99  394 
9.  99  419 
9.  99  444 
9.  99  460 
9.  99  495 
9.99  520 
9.  99  545 
9. 99  570 
9.99  596 
9.  99  621 
9.  99  646 
9.99  672 
9.  99  697 
9.  99  722 
9.99  747 
9.99  773 
9.  99  798 
9.99  823 
9.  99  848 
9.  99  874 
9.  99  899 
9.99  924 
9.  99  949 
9.  99  975 


L.  Cotg. 


0.  01  516 
0.  01  491 
0.  01  460 
0.  01  440 
0.01  415 


0.01  390 
0. 01  365 
0.01  339 
0.  01  314 
0.  01  289 


0. 01  263 
0.01  238 
0.01  213 
0.  01  188 
0.  01  162 


0.01  137 
0.01  112 
0.  01  087 
0.  01  061 
0.  01  036 


0.01  Oil 
0.00  985 
0.00  960 
0.00  935 
0.  00  910 
0.  00  884 
0.00  859 
0.  00  834 
0.00  809 
0. 00  783 
0.00  758 
0.00  733 
0. GO  707 
0.00  682 


0.  ( 


657 


0.  00  632 
0.  00  606 
0.  00  581 
0.00  556 
0.  00  .531 
0. 00  505 
0.  00  480 
0. 00  455 
0.00  430 
0.  00  404 
0. 00  379 
0.  00  354 
0. 00  328 
0.00  303 
0.  00  278 
0, 00  253 
0.  00  227 
0.  00  202 
0.  00  177 
0.00  152 
0.  00  126 
0.00  101 
0.  00  076 
0.  00  051 
0.00  025 
0.  00  000 


9.85  669 
9.  85  657 
9.85  645^ 
9. 85  632 
9.85  620 
9. 85  608 
3.85  596 
9.85  583 


9. 85  571 
9.85  559 
9. 85  547 
9.85  534 
9. 85  522 


9. 85  510 
9.85  497 
9.  85  485 
9. 85  473 
9.  85  460 


9. 85  448 
9.85  436 
9.  85  423 
9.85  411 
9.85  399 
9.  85  386 
9. 85  374 
9.85  361 
9.85  349 
9.85  337 
9.85  324 
9. 85  312 
9.85  299 
9.85  287 
9. 85  274 
9.85  262 
9.85  2.50 
9. 85  237 
9.  85  225 
9. 85  212 
9.85  200 
9.85  187 
9.85  175 
9.  85  162 
9.85  150 
9.85  137 
9. 85  125 
9.85  112 
9.85  100 
9.85  087 
9. 85  074 
9.85  062 
9.  85  049 
9.85  037 
9. 85  024 


0,4 
0,9 
1,3 
h^ 
2,2 
2,6 
3,0 
3,5 
3,9 
4,3 
8,7 
13,0 
17,3 
21,7 


1,7 
2,1 
2,5 
2,9 
3,3 
3,8 
4,2 
8,3 
12,5 
16,7 
20,8 


14 

13 

0,2 

0,2 

0,5 

0,4 

0,7 

0,6 

0,9 

0,9 

1,2 

1,1 

1,4 

1,3 

1,6 

1,5 

1,9 

1,7 

2,1 

•2,0 

2,3 

2,2 

4,7 

4,3 

7,0 

6,5 

9,3 

8,7 

11,7 

10,8 

3,0 
5,0 
7,0 
9,0 
11,0 
13,0 
15,0 
17,0 
19,0 
21,0 
23,0 
25,0 


1,1 
3,2 
5,4 
7,6 
9,8 
11,9 
14,1 
16,2 
18,4 
20,6 
22,8 
24,9 


0,9 
2,9 
4,8 
6,7 
8,7 
10,6 
12,5 
14,4 
16,3 
18,3 
20,2 
23,1 
24,1 


1,0 
3,1 
5,2 
7,3 
9,4 
11,5 
13,5 
15,6 
17,7 
19,8 
21,9 
23,9 


45= 


INDEX 


Acciiracy  of  control 

Adirondack  survey 

Alidade 

for  traversing 

Altitudes,  measurement  of,  in  connection  with  traverse 

linea 

"witli  plane  table 

Amount  of  control 

Amphitheaters  

Aneroid 

Apparent  time 

Aqueous  agencies 

Arid  region,  erosion  in 

Astronomic  determination  of  position 

Astronomical  station,  selection  of 

transit  and  zenith  telescope 

A  zimuth,  correction  for  deviation  in 

observations,  example  of  record 

example  of  reduction 

tor 

on  Polaris  at  elongation 

reduction  of 

summary  of  results 

Baldwin  device  for  stretching  tape  in  base  line  meas- 
urement  

Barometric  observations,  reduction  of 

tables,  use  of 

Base  level 

line,  alignment  of. 

measurement 

example  of  reduction  of 

instruments  used  in 

personnel  of  party 

reduction  of 

tension  of  tape  in 

selection  of  site 

Batteries  in  use 

Canyons,  formation  of 

in  strata,  alternating  hard  and  soft 

Chronograph 

Chronometer,  break  circuit 

Cistern  barometer 

filling  of  tubes 

method  of  use 

Classification  of  work 

Coast  and  Geodetic  Survey,  United  States 

Collimation,  correction  for  error  of 

Colors  used  on  original  maps 

Comparison  of  time 

Contour  interval 

Conventions 

Corrasion 

Declination 


Declinations,  apparent,  computationof 

Deposition  from  volcanic  action 

water 

the  atmosphere 

Disintegration 

Distances,  computation  of 

Diurnal  aberration,  correction  for 

Douglas  odometer 

Erosion 

European  maps,  scales  of 

Features  represented 

Field  work  of  astronomical  determination 

scale  of 

Figure  adjustment 

Fortieth  parallel  survey 

Greneralization  of  maps 

Geodetic  coordinates 

Geological  and  Geographical  Survey  of  Territories 

Geometric  control 

Glacial  deposition 

Heliotrope,  Coast  Survey  form 

Steinheil 

Horizontal  angles,  errors  incident  to  measurement  of . . 

form  of  record 

instructions  for  measurement  of 

order  of  readings 

location 

Inequality  of  pivots,  correction  for 

Inspection 

Introduction 

Johnson  plane  table 

Lake  survey,  United  States 

Land  OflBce  plats 

surveys  

Latitude  determination,  form  of  record  of 

how  determined 

observations 

list  of  stars  for 

reduction  of 

Least  squares  in  figure  adjustment 

station  adjustment 

Legencls  upon  maps 

Level,  corrections  for  error  of 

division,  measurement  of 

Longitude  determination,  example  of  reduction 

how  determined - 

Massachusetts,  Borden  survey  of 

Mean  time - 

Method  of  adjusting  transit  in  meridian - 

control 

Micrometer  screw,  measurement  of  division  of  head  of 

New  Jersey  State  survey 

299 


300 


INDEX. 


Page. 

New  York  State  survey ^ 

Nortliern  trauscontiueutnl  survey 3 

Odometers ^^ 

Offloework 128 

Organization  of  parties *1 

Pennsylvania  State  survey 5 

Personal  equation 35 

Piracy H^ 

Plan  of  map  of  United  States 6 

Plane  table ^ 

sheets 82 

Primary  elevations '''' 

triangulation -IS 

prosecution  of  work 63 

selection  of  stations 49 

Private  surveys 5 

Profiles  of  streams 112 

Projections 129 

Public  land  surveys,  plan  of 101 

utilization  of 101 

Eailroad  profiles 6 

surveys 5 

Eeduction  to  center 65 

Reports 125 

Eight  ascension 17 

Eocky  Mountain  region,  survey  of 3 

Scaleof  United  States  map 7 

Secondary  triangulation 79 

Sidereal  time 17 

Signals  and  observing  towers  in  triangulation 51 

in  triangulation 50 

Sinks,  origin  of 115 


Size  of  sheets 10 

Sketching U-106 

Solar  time 17 

Spherical  excess 65 

Stadia  measurement ^ 92 

Station  error 35 

Station  adjustment 66 

Support  for  astronomical  transit 21 

Surveys  under  United  States  Government 2 

Talcott's  method 17 

Theodolites  for  triangulation 5i 

Three-point  problem 83 

Time  determination,  example  of  record 32 

observations  for 28 

reduction  of 29 

Titles  of  maps 130 

Topographic  features,  origin  of 108 

forms,  influence  of  structure  upon 117 

parties,  distribution  of  work  in 91 

Transportation HI 

Traversing 12, 13 

Traverse  lines  for  primary  control 75 

work 85 

plane  tables  for 86 

Triangulation 12 

Uplift 108 

Water  gaps 116 

Weathering HI 

Wind  gaps 116 

Zenith  distance 17 

telescope  and  asti'!>uomical  transit -  -  18 


\ 


\ 


k 


y.^}^^ 


/. 


LEGEND 

1  I  NorOuSTi  I'naAc  Ttaiui.  Survey. 

^^m^p  I'-^.t'otttit  anil  Oeodetiv  Sut^fty. 


\Cf 


\       ■*' 


^ 


t 

Vi 

\^?^ 

•^    / 

1 

c^\ 

\ 

AJ/ 

<^ 

.- 

• 

•k    \ 

<* 

^""'  '.     . 

' 

\     '  '' 

g&-: 


it^i 


^      ***-'-N®V-5^ 


^ 


siioinxn  I'lioGUKSS  OFTKLVNT.i-LVTiox  Tomciui'in- 

ASTRONOMIC  LOC.ATKIX. 


^ 


■y^, 


